Real Analysis - Revision history

Wordpad: BloomWiki: Real Analysis

2026-04-25T01:56:52Z

BloomWiki: Real Analysis

← Older revision		Revision as of 01:56, 25 April 2026
Line 1:		Line 1:
			<div style="background-color: #4B0082; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
	{{BloomIntro}}		{{BloomIntro}}
	Real analysis is the rigorous mathematical theory underlying calculus. Where calculus courses teach you how to differentiate and integrate functions, real analysis asks ''why'' those procedures work: what does it mean for a function to be continuous, differentiable, or integrable? What is the precise meaning of a limit? When can we interchange limits and integrals? Real analysis resolves the conceptual puzzles that caused 200 years of confusion after Newton and Leibniz invented calculus. Its foundational concept — the epsilon-delta definition of limits — provides the logical bedrock for all of analysis, and its theorems (the Heine-Cantor theorem, the mean value theorem, Lebesgue's dominated convergence theorem) underpin every branch of mathematics that uses continuous quantities.		Real analysis is the rigorous mathematical theory underlying calculus. Where calculus courses teach you how to differentiate and integrate functions, real analysis asks ''why'' those procedures work: what does it mean for a function to be continuous, differentiable, or integrable? What is the precise meaning of a limit? When can we interchange limits and integrals? Real analysis resolves the conceptual puzzles that caused 200 years of confusion after Newton and Leibniz invented calculus. Its foundational concept — the epsilon-delta definition of limits — provides the logical bedrock for all of analysis, and its theorems (the Heine-Cantor theorem, the mean value theorem, Lebesgue's dominated convergence theorem) underpin every branch of mathematics that uses continuous quantities.
			</div>

	== Remembering ==		__TOC__

			<div style="background-color: #000080; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Remembering</span> ==
	* '''Real analysis''' — The rigorous mathematical study of the real numbers, sequences, series, limits, continuity, differentiation, and integration.		* '''Real analysis''' — The rigorous mathematical study of the real numbers, sequences, series, limits, continuity, differentiation, and integration.
	* '''Real numbers (ℝ)''' — The complete ordered field; characterized uniquely (up to isomorphism) by the least upper bound property.		* '''Real numbers (ℝ)''' — The complete ordered field; characterized uniquely (up to isomorphism) by the least upper bound property.
Line 18:		Line 23:
	* '''Uniform convergence''' — A sequence of functions fₙ → f uniformly if: ∀ε>0, ∃N: ∀x, n>N ⟹ \|fₙ(x)-f(x)\|<ε.		* '''Uniform convergence''' — A sequence of functions fₙ → f uniformly if: ∀ε>0, ∃N: ∀x, n>N ⟹ \|fₙ(x)-f(x)\|<ε.
	* '''Power series''' — A series Σ aₙxⁿ; converges in a disk; the basis of Taylor series.		* '''Power series''' — A series Σ aₙxⁿ; converges in a disk; the basis of Taylor series.
			</div>

	== Understanding ==		<div style="background-color: #006400; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Understanding</span> ==
	Real analysis develops four fundamental concepts with increasing rigor:		Real analysis develops four fundamental concepts with increasing rigor:

Line 29:		Line 36:

	'''Series and the problem of interchanging limits''': When can we integrate or differentiate a series term-by-term? Uniform convergence is the key: if fₙ → f uniformly and each fₙ is continuous (resp. integrable), then f is continuous (resp. integrable) and ∫fₙ → ∫f. Point-wise convergence is not enough: the classic example of functions fₙ(x) = xⁿ on [0,1] converges pointwise to a discontinuous function. Uniform convergence distinguishes safe limit-interchange from dangerous ones.		'''Series and the problem of interchanging limits''': When can we integrate or differentiate a series term-by-term? Uniform convergence is the key: if fₙ → f uniformly and each fₙ is continuous (resp. integrable), then f is continuous (resp. integrable) and ∫fₙ → ∫f. Point-wise convergence is not enough: the classic example of functions fₙ(x) = xⁿ on [0,1] converges pointwise to a discontinuous function. Uniform convergence distinguishes safe limit-interchange from dangerous ones.
			</div>

	== Applying ==		<div style="background-color: #8B0000; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Applying</span> ==
	'''Implementing fundamental analysis concepts:'''		'''Implementing fundamental analysis concepts:'''
	<syntaxhighlight lang="python">		<syntaxhighlight lang="python">
Line 109:		Line 118:
	: '''Lebesgue integration''' → Lebesgue (''Leçons sur l'intégration'', 1904)		: '''Lebesgue integration''' → Lebesgue (''Leçons sur l'intégration'', 1904)
	: '''Functional analysis''' → Banach, Hilbert — analysis on infinite-dimensional spaces		: '''Functional analysis''' → Banach, Hilbert — analysis on infinite-dimensional spaces
			</div>

	== Analyzing ==		<div style="background-color: #8B4500; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Analyzing</span> ==
	{\| class="wikitable"		{\| class="wikitable"
	\|+ Key Theorems of Real Analysis		\|+ Key Theorems of Real Analysis
Line 129:		Line 140:

	'''Subtleties and counterexamples''': Pointwise vs. uniform convergence: fₙ(x) = xⁿ on [0,1] converges pointwise but not uniformly (limit is discontinuous). Weierstrass function: continuous everywhere, differentiable nowhere — shatters the intuition that continuity implies differentiability. The Devil's Staircase (Cantor function): monotone, continuous, and yet its derivative is zero almost everywhere while it still climbs from 0 to 1.		'''Subtleties and counterexamples''': Pointwise vs. uniform convergence: fₙ(x) = xⁿ on [0,1] converges pointwise but not uniformly (limit is discontinuous). Weierstrass function: continuous everywhere, differentiable nowhere — shatters the intuition that continuity implies differentiability. The Devil's Staircase (Cantor function): monotone, continuous, and yet its derivative is zero almost everywhere while it still climbs from 0 to 1.
			</div>

	== Evaluating ==		<div style="background-color: #483D8B; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Evaluating</span> ==
	Real analysis proofs are evaluated by:		Real analysis proofs are evaluated by:
	# '''Logical rigor''': are all epsilon-delta arguments formally correct?		# '''Logical rigor''': are all epsilon-delta arguments formally correct?
Line 137:		Line 150:
	# '''Counterexamples''': for each condition, is there a counterexample showing it's necessary?		# '''Counterexamples''': for each condition, is there a counterexample showing it's necessary?
	# '''Generalizability''': does the result extend to metric spaces, Banach spaces, or abstract measure spaces?		# '''Generalizability''': does the result extend to metric spaces, Banach spaces, or abstract measure spaces?
			</div>

	== Creating ==		<div style="background-color: #2F4F4F; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
			== <span style="color: #FFFFFF;">Creating</span> ==
	Advanced analysis directions:		Advanced analysis directions:
	# '''Measure theory and Lebesgue integration''': define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem.		# '''Measure theory and Lebesgue integration''': define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem.
Line 149:		Line 164:
	[[Category:Real Analysis]]		[[Category:Real Analysis]]
	]]		]]
			</div>

Wordpad: BloomWiki: Real Analysis

2026-04-23T14:36:42Z

BloomWiki: Real Analysis

← Older revision		Revision as of 14:36, 23 April 2026
Line 131:		Line 131:

	== Evaluating ==		== Evaluating ==
	Real analysis proofs are evaluated by: ~~(1)~~ '''Logical rigor''': are all epsilon-delta arguments formally correct? ~~(2)~~ '''Necessity of hypotheses''': can any condition be weakened? ~~(3)~~ '''Constructiveness''': does the proof exhibit the claimed object or merely show it exists? ~~(4)~~ '''Counterexamples''': for each condition, is there a counterexample showing it's necessary? ~~(5)~~ '''Generalizability''': does the result extend to metric spaces, Banach spaces, or abstract measure spaces?		Real analysis proofs are evaluated by:
			# '''Logical rigor''': are all epsilon-delta arguments formally correct?
			# '''Necessity of hypotheses''': can any condition be weakened?
			# '''Constructiveness''': does the proof exhibit the claimed object or merely show it exists?
			# '''Counterexamples''': for each condition, is there a counterexample showing it's necessary?
			# '''Generalizability''': does the result extend to metric spaces, Banach spaces, or abstract measure spaces?

	== Creating ==		== Creating ==
	Advanced analysis directions: ~~(1)~~ '''Measure theory and Lebesgue integration''': define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem. ~~(2)~~ '''Functional analysis''': study spaces of functions (L², L^p, Sobolev spaces) as infinite-dimensional vector spaces; Hilbert spaces, Banach spaces, bounded linear operators. ~~(3)~~ '''Complex analysis''': analysis over ℂ; holomorphic functions, Cauchy's theorem, conformal maps — remarkably richer than real analysis. ~~(4)~~ '''Fourier analysis''': decompose functions into frequency components; Fourier series, transforms, and their applications in differential equations and signal processing. ~~(5)~~ '''Ordinary and partial differential equations''': real analysis provides the existence-uniqueness theorems (Picard-Lindelöf) and regularity theory.		Advanced analysis directions:
			# '''Measure theory and Lebesgue integration''': define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem.
			# '''Functional analysis''': study spaces of functions (L², L^p, Sobolev spaces) as infinite-dimensional vector spaces; Hilbert spaces, Banach spaces, bounded linear operators.
			# '''Complex analysis''': analysis over ℂ; holomorphic functions, Cauchy's theorem, conformal maps — remarkably richer than real analysis.
			# '''Fourier analysis''': decompose functions into frequency components; Fourier series, transforms, and their applications in differential equations and signal processing.
			# '''Ordinary and partial differential equations''': real analysis provides the existence-uniqueness theorems (Picard-Lindelöf) and regularity theory.

	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Real Analysis]]		[[Category:Real Analysis]]
	]]		]]

Wordpad: BloomWiki: Real Analysis

2026-04-23T14:21:28Z

BloomWiki: Real Analysis

← Older revision		Revision as of 14:21, 23 April 2026
Line 22:		Line 22:
	Real analysis develops four fundamental concepts with increasing rigor:		Real analysis develops four fundamental concepts with increasing rigor:

	Limits and the completeness of ℝ: The rationals ℚ have "holes" — the sequence 1, 1.4, 1.41, 1.414, ... converges to √2, which isn't rational. The reals are constructed to fill these holes: Dedekind cuts or Cauchy completion. The least upper bound property (every bounded-above non-empty set has a supremum) is the key axiom. Consequences: the Bolzano-Weierstrass theorem (every bounded sequence in ℝ has a convergent subsequence), the Heine-Borel theorem (a subset of ℝ^n is compact iff it is closed and bounded).		'''Limits and the completeness of ℝ''': The rationals ℚ have "holes" — the sequence 1, 1.4, 1.41, 1.414, ... converges to √2, which isn't rational. The reals are constructed to fill these holes: Dedekind cuts or Cauchy completion. The least upper bound property (every bounded-above non-empty set has a supremum) is the key axiom. Consequences: the Bolzano-Weierstrass theorem (every bounded sequence in ℝ has a convergent subsequence), the Heine-Borel theorem (a subset of ℝ^n is compact iff it is closed and bounded).

	Continuity and its consequences: The epsilon-delta definition captures the intuition that a function is continuous if nearby inputs give nearby outputs, without any reference to "infinitesimals." Key theorems: the Extreme Value Theorem (a continuous function on a compact set attains its max and min — used in optimization), the Intermediate Value Theorem (a continuous function on [a,b] that takes values c₁ and c₂ takes all values between them — used to prove the existence of roots), and Uniform Continuity on compact sets.		'''Continuity and its consequences''': The epsilon-delta definition captures the intuition that a function is continuous if nearby inputs give nearby outputs, without any reference to "infinitesimals." Key theorems: the Extreme Value Theorem (a continuous function on a compact set attains its max and min — used in optimization), the Intermediate Value Theorem (a continuous function on [a,b] that takes values c₁ and c₂ takes all values between them — used to prove the existence of roots), and Uniform Continuity on compact sets.

	Differentiation and integration: The Fundamental Theorem of Calculus (FTC) is the deepest result of basic analysis: differentiation and integration are inverse operations. But the theorem requires precise formulation: f must be integrable, F = ∫ₐˣ f(t)dt is differentiable with F' = f (FTC Part 1); if F' = f on [a,b] then ∫ₐᵇ f = F(b) - F(a) (FTC Part 2). The mean value theorem (f(b)-f(a) = f'(c)(b-a) for some c in (a,b)) underlies error bounds for approximations.		'''Differentiation and integration''': The Fundamental Theorem of Calculus (FTC) is the deepest result of basic analysis: differentiation and integration are inverse operations. But the theorem requires precise formulation: f must be integrable, F = ∫ₐˣ f(t)dt is differentiable with F' = f (FTC Part 1); if F' = f on [a,b] then ∫ₐᵇ f = F(b) - F(a) (FTC Part 2). The mean value theorem (f(b)-f(a) = f'(c)(b-a) for some c in (a,b)) underlies error bounds for approximations.

	Series and the problem of interchanging limits: When can we integrate or differentiate a series term-by-term? Uniform convergence is the key: if fₙ → f uniformly and each fₙ is continuous (resp. integrable), then f is continuous (resp. integrable) and ∫fₙ → ∫f. Point-wise convergence is not enough: the classic example of functions fₙ(x) = xⁿ on [0,1] converges pointwise to a discontinuous function. Uniform convergence distinguishes safe limit-interchange from dangerous ones.		'''Series and the problem of interchanging limits''': When can we integrate or differentiate a series term-by-term? Uniform convergence is the key: if fₙ → f uniformly and each fₙ is continuous (resp. integrable), then f is continuous (resp. integrable) and ∫fₙ → ∫f. Point-wise convergence is not enough: the classic example of functions fₙ(x) = xⁿ on [0,1] converges pointwise to a discontinuous function. Uniform convergence distinguishes safe limit-interchange from dangerous ones.

	== Applying ==		== Applying ==
Line 131:		Line 131:

	== Evaluating ==		== Evaluating ==
	Real analysis proofs are evaluated by: (1) Logical rigor: are all epsilon-delta arguments formally correct? (2) Necessity of hypotheses: can any condition be weakened? (3) Constructiveness: does the proof exhibit the claimed object or merely show it exists? (4) Counterexamples: for each condition, is there a counterexample showing it's necessary? (5) Generalizability: does the result extend to metric spaces, Banach spaces, or abstract measure spaces?		Real analysis proofs are evaluated by: (1) '''Logical rigor''': are all epsilon-delta arguments formally correct? (2) '''Necessity of hypotheses''': can any condition be weakened? (3) '''Constructiveness''': does the proof exhibit the claimed object or merely show it exists? (4) '''Counterexamples''': for each condition, is there a counterexample showing it's necessary? (5) '''Generalizability''': does the result extend to metric spaces, Banach spaces, or abstract measure spaces?

	== Creating ==		== Creating ==
	Advanced analysis directions: (1) Measure theory and Lebesgue integration: define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem. (2) Functional analysis: study spaces of functions (L², L^p, Sobolev spaces) as infinite-dimensional vector spaces; Hilbert spaces, Banach spaces, bounded linear operators. (3) Complex analysis: analysis over ℂ; holomorphic functions, Cauchy's theorem, conformal maps — remarkably richer than real analysis. (4) Fourier analysis: decompose functions into frequency components; Fourier series, transforms, and their applications in differential equations and signal processing. (5) Ordinary and partial differential equations: real analysis provides the existence-uniqueness theorems (Picard-Lindelöf) and regularity theory.		Advanced analysis directions: (1) '''Measure theory and Lebesgue integration''': define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem. (2) '''Functional analysis''': study spaces of functions (L², L^p, Sobolev spaces) as infinite-dimensional vector spaces; Hilbert spaces, Banach spaces, bounded linear operators. (3) '''Complex analysis''': analysis over ℂ; holomorphic functions, Cauchy's theorem, conformal maps — remarkably richer than real analysis. (4) '''Fourier analysis''': decompose functions into frequency components; Fourier series, transforms, and their applications in differential equations and signal processing. (5) '''Ordinary and partial differential equations''': real analysis provides the existence-uniqueness theorems (Picard-Lindelöf) and regularity theory.

	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Real Analysis]]		[[Category:Real Analysis]]
	]]		]]

Wordpad: BloomWiki: Real Analysis

2026-04-23T13:09:09Z

BloomWiki: Real Analysis

New page

{{BloomIntro}}
Real analysis is the rigorous mathematical theory underlying calculus. Where calculus courses teach you how to differentiate and integrate functions, real analysis asks ''why'' those procedures work: what does it mean for a function to be continuous, differentiable, or integrable? What is the precise meaning of a limit? When can we interchange limits and integrals? Real analysis resolves the conceptual puzzles that caused 200 years of confusion after Newton and Leibniz invented calculus. Its foundational concept — the epsilon-delta definition of limits — provides the logical bedrock for all of analysis, and its theorems (the Heine-Cantor theorem, the mean value theorem, Lebesgue's dominated convergence theorem) underpin every branch of mathematics that uses continuous quantities.

== Remembering ==
* '''Real analysis''' — The rigorous mathematical study of the real numbers, sequences, series, limits, continuity, differentiation, and integration.
* '''Real numbers (ℝ)''' — The complete ordered field; characterized uniquely (up to isomorphism) by the least upper bound property.
* '''Least upper bound (supremum)''' — The smallest upper bound of a set; the completeness axiom: every non-empty set with an upper bound has a supremum in ℝ.
* '''Completeness of ℝ''' — Every Cauchy sequence in ℝ converges (to a real number); no "holes" in ℝ (unlike ℚ).
* '''Sequence''' — A function ℕ → ℝ; the foundation of limit theory.
* '''Limit of a sequence''' — L is the limit of (aₙ) if: for every ε > 0, ∃ N such that n > N ⟹ |aₙ - L| < ε.
* '''Cauchy sequence''' — A sequence where |aₙ - aₘ| → 0 as n,m → ∞; converges iff ℝ is complete.
* '''Epsilon-delta definition (continuity)''' — f is continuous at c if: ∀ε>0, ∃δ>0: |x-c|<δ ⟹ |f(x)-f(c)|<ε.
* '''Uniform continuity''' — δ depends only on ε, not on the point c; stronger than pointwise continuity.
* '''Differentiability''' — f'(x) = lim_{h→0} [f(x+h)-f(x)]/h; requires the limit to exist.
* '''Riemann integral''' — Defined via upper and lower sums; the standard calculus integral.
* '''Lebesgue integral''' — A more powerful integral based on measure theory; integrates functions Riemann cannot.
* '''Series''' — Sum of a sequence; Σaₙ; convergence is the limit of partial sums.
* '''Uniform convergence''' — A sequence of functions fₙ → f uniformly if: ∀ε>0, ∃N: ∀x, n>N ⟹ |fₙ(x)-f(x)|<ε.
* '''Power series''' — A series Σ aₙxⁿ; converges in a disk; the basis of Taylor series.

== Understanding ==
Real analysis develops four fundamental concepts with increasing rigor:

**Limits and the completeness of ℝ**: The rationals ℚ have "holes" — the sequence 1, 1.4, 1.41, 1.414, ... converges to √2, which isn't rational. The reals are constructed to fill these holes: Dedekind cuts or Cauchy completion. The least upper bound property (every bounded-above non-empty set has a supremum) is the key axiom. Consequences: the Bolzano-Weierstrass theorem (every bounded sequence in ℝ has a convergent subsequence), the Heine-Borel theorem (a subset of ℝ^n is compact iff it is closed and bounded).

**Continuity and its consequences**: The epsilon-delta definition captures the intuition that a function is continuous if nearby inputs give nearby outputs, without any reference to "infinitesimals." Key theorems: the Extreme Value Theorem (a continuous function on a compact set attains its max and min — used in optimization), the Intermediate Value Theorem (a continuous function on [a,b] that takes values c₁ and c₂ takes all values between them — used to prove the existence of roots), and Uniform Continuity on compact sets.

**Differentiation and integration**: The Fundamental Theorem of Calculus (FTC) is the deepest result of basic analysis: differentiation and integration are inverse operations. But the theorem requires precise formulation: f must be integrable, F = ∫ₐˣ f(t)dt is differentiable with F' = f (FTC Part 1); if F' = f on [a,b] then ∫ₐᵇ f = F(b) - F(a) (FTC Part 2). The mean value theorem (f(b)-f(a) = f'(c)(b-a) for some c in (a,b)) underlies error bounds for approximations.

**Series and the problem of interchanging limits**: When can we integrate or differentiate a series term-by-term? Uniform convergence is the key: if fₙ → f uniformly and each fₙ is continuous (resp. integrable), then f is continuous (resp. integrable) and ∫fₙ → ∫f. Point-wise convergence is not enough: the classic example of functions fₙ(x) = xⁿ on [0,1] converges pointwise to a discontinuous function. Uniform convergence distinguishes safe limit-interchange from dangerous ones.

== Applying ==
'''Implementing fundamental analysis concepts:'''
<syntaxhighlight lang="python">
import math
from typing import Callable, Iterator

# === Epsilon-Delta Verification ===
def verify_continuity_at(f: Callable[[float], float], c: float, epsilon: float,
test_delta: float, n_tests: int = 1000) -> dict:
"""
Empirically search for a δ > 0 such that |x-c| < δ ⟹ |f(x)-f(c)| < ε.
Illustrates the ε-δ definition (not a formal proof).
"""
fc = f(c)
deltas_tried = [test_delta * (0.1 ** k) for k in range(5)]
for delta in deltas_tried:
# Test n_tests points in (c-δ, c+δ)
xs = [c + delta * (2*i/(n_tests-1) - 1) for i in range(n_tests)]
violations = [x for x in xs if abs(x-c) < delta and abs(f(x)-fc) >= epsilon]
if not violations:
return {'epsilon': epsilon, 'delta_found': delta, 'continuous_at': c,
'verdict': 'Likely continuous (no violations found)'}
return {'epsilon': epsilon, 'verdict': 'Violations found — not continuous or need smaller delta'}

# Test continuity of f(x) = x² at c=2
result = verify_continuity_at(lambda x: x**2, c=2.0, epsilon=0.1, test_delta=0.01)
print(result)

# === Sequence Convergence ===
def check_convergence(seq: Iterator[float], limit: float, epsilon: float,
max_n: int = 10000) -> dict:
"""Find N such that n > N ⟹ |aₙ - L| < ε."""
for n, an in enumerate(seq):
if n > max_n: return {'converges': False, 'N_found': None}
if abs(an - limit) < epsilon:
return {'converges': True, 'N': n, 'value_at_N': an}
return {'converges': False}

# aₙ = 1/n → 0
one_over_n = (1/n for n in range(1, 100001))
print(check_convergence(one_over_n, limit=0, epsilon=0.01))

# === Taylor Series with Remainder ===
def taylor_exp(x: float, n_terms: int) -> tuple[float, float]:
"""Approximate eˣ with n-term Taylor series; compute Lagrange remainder."""
approx = sum(x**k / math.factorial(k) for k in range(n_terms))
# Lagrange remainder: R_n = eˢ * x^n / n! for some s between 0 and x
max_s = abs(x)
remainder_bound = math.exp(max_s) * abs(x)**n_terms / math.factorial(n_terms)
return approx, remainder_bound

for n in [3, 5, 10, 15]:
approx, bound = taylor_exp(1.0, n)
print(f"n={n}: e≈{approx:.10f}, error≤{bound:.2e}, actual error={abs(approx-math.e):.2e}")

# === Riemann Integration ===
def riemann_integral(f: Callable[[float], float], a: float, b: float, n: int,
method='midpoint') -> float:
"""Approximate ∫ₐᵇ f using n-point Riemann sum."""
dx = (b - a) / n
if method == 'left':
xs = [a + i*dx for i in range(n)]
elif method == 'right':
xs = [a + (i+1)*dx for i in range(n)]
else: # midpoint
xs = [a + (i + 0.5)*dx for i in range(n)]
return sum(f(x) * dx for x in xs)

# Verify FTC: ∫₀¹ x² dx = [x³/3]₀¹ = 1/3
for n in [10, 100, 1000, 10000]:
approx = riemann_integral(lambda x: x**2, 0, 1, n)
print(f"n={n}: ∫x²dx≈{approx:.6f}, error={abs(approx-1/3):.2e}")
</syntaxhighlight>

; Key texts and theorists
: '''Foundational''' → Cauchy (rigorized limits), Weierstrass (ε-δ definition), Dedekind (real number construction)
: '''Classic textbooks''' → Rudin (''Principles of Mathematical Analysis''), Apostol, Bartle
: '''Lebesgue integration''' → Lebesgue (''Leçons sur l'intégration'', 1904)
: '''Functional analysis''' → Banach, Hilbert — analysis on infinite-dimensional spaces

== Analyzing ==
{| class="wikitable"
|+ Key Theorems of Real Analysis
! Theorem !! Statement !! Why It Matters
|-
| Least Upper Bound || Every bounded non-empty set has a supremum in ℝ || The defining property of ℝ; proves everything else
|-
| Bolzano-Weierstrass || Every bounded sequence has a convergent subsequence || Foundation of compactness arguments
|-
| Extreme Value Theorem || Continuous f on compact K attains max and min || Guarantees optima exist
|-
| Intermediate Value Theorem || Continuous f on [a,b]; c between f(a) and f(b) → ∃x: f(x)=c || Proves existence of roots; fixed points
|-
| Mean Value Theorem || f differentiable on (a,b) → ∃c: f'(c)=(f(b)-f(a))/(b-a) || Foundation of Taylor bounds, monotonicity
|-
| FTC || Differentiation and integration are inverses || Converts area problems to antiderivatives
|}

'''Subtleties and counterexamples''': Pointwise vs. uniform convergence: fₙ(x) = xⁿ on [0,1] converges pointwise but not uniformly (limit is discontinuous). Weierstrass function: continuous everywhere, differentiable nowhere — shatters the intuition that continuity implies differentiability. The Devil's Staircase (Cantor function): monotone, continuous, and yet its derivative is zero almost everywhere while it still climbs from 0 to 1.

== Evaluating ==
Real analysis proofs are evaluated by: (1) **Logical rigor**: are all epsilon-delta arguments formally correct? (2) **Necessity of hypotheses**: can any condition be weakened? (3) **Constructiveness**: does the proof exhibit the claimed object or merely show it exists? (4) **Counterexamples**: for each condition, is there a counterexample showing it's necessary? (5) **Generalizability**: does the result extend to metric spaces, Banach spaces, or abstract measure spaces?

== Creating ==
Advanced analysis directions: (1) **Measure theory and Lebesgue integration**: define integration for a vastly broader class of functions; prove dominated convergence, monotone convergence, Fubini's theorem. (2) **Functional analysis**: study spaces of functions (L², L^p, Sobolev spaces) as infinite-dimensional vector spaces; Hilbert spaces, Banach spaces, bounded linear operators. (3) **Complex analysis**: analysis over ℂ; holomorphic functions, Cauchy's theorem, conformal maps — remarkably richer than real analysis. (4) **Fourier analysis**: decompose functions into frequency components; Fourier series, transforms, and their applications in differential equations and signal processing. (5) **Ordinary and partial differential equations**: real analysis provides the existence-uniqueness theorems (Picard-Lindelöf) and regularity theory.

[[Category:Mathematics]]
[[Category:Real Analysis]]
]]