Editing Real Analysis

<div style="background-color: #4B0082; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
{{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.
</div>

__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 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.
</div>

<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:

'''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.
</div>

<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:'''
<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
</div>

<div style="background-color: #8B4500; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
== <span style="color: #FFFFFF;">Analyzing</span> ==
{| 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.
</div>

<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:
# '''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?
</div>

<div style="background-color: #2F4F4F; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">
== <span style="color: #FFFFFF;">Creating</span> ==
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:Real Analysis]]
]]
</div>