List of Non-Examples in Analysis

Introduction

This article lists the non-examples of analysis that the corpus records: the objects that are defined by the same words as a standard positive object and fail one of its properties. Each row names the non-example, records the positive object it fails to be and the failure itself, and points to the article that introduces the non-example. Every row points to an article; this article introduces nothing and proves nothing. The non-examples are grouped by the layer that introduces them: the five standard counterexamples of the real line and its measure theory, the further non-examples of measure and integration, the non-examples of the convergence of functions, the non-examples of differentiation and analyticity, and the non-examples of series and integrals. The positive object that each non-example fails to be is named beside it, so that the reader sees the example and the non-example side by side.

The Standard Counterexamples

Non-example	The positive object it fails to be, and the failure	Introduced in
the continuous nowhere differentiable function	a differentiable function: the function is continuous everywhere and has no derivative at any point. The corpus records the existence of such functions by the Baire category theorem, in the meagre-set argument, rather than by the classical Weierstrass series, which is not introduced in Parts I to III	Metric, Uniform and Complete Spaces
the path of Brownian motion	a differentiable function: the path is almost surely continuous and almost surely nowhere differentiable, with infinite variation	Brownian Motion and Stochastic Calculus
Volterra's function	a function whose derivative is Riemann integrable: it is differentiable, its derivative is bounded, and the derivative is not Riemann integrable, being discontinuous on a fat Cantor set of positive measure, so the fundamental theorem of calculus fails without the continuity of the derivative	Measure Theory and Integration
the Dirichlet function $\mathbf 1_{\mathbb Q}$	a Riemann integrable function: it is bounded and Lebesgue integrable with integral $0$, the rationals being null, and it is not Riemann integrable, being discontinuous everywhere	Real Integration; Measure Theory and Integration
a non-measurable set, the Vitali set	a Lebesgue measurable set: it has no length consistent with translation invariance and countable additivity, so it lies outside the Lebesgue $\sigma$-algebra	Measure Theory and Integration
the smooth function $\exp(-1/x^2)$ extended by $0$ at the origin	an analytic function: all its derivatives at $0$ vanish, so its Taylor series is the zero series, while the function is positive away from $0$	Real Analysis

Non-Examples of Measure and Integration

Non-example	The positive object it fails to be, and the failure	Introduced in
the Cantor function	an absolutely continuous function: it is continuous, nondecreasing and constant on each complement of the Cantor set, with derivative $0$ almost everywhere, yet it rises from $0$ to $1$	Measure Theory and Integration; Measure-Theoretic Probability
the Cantor distribution	an absolutely continuous law: it is singular continuous, with no density, and its distribution function is the Cantor function	Measure-Theoretic Probability
the indicator of a fat Cantor set	a Riemann integrable function: it is Lebesgue integrable and discontinuous on a set of positive measure	Real Integration
the counting measure on an uncountable set	a $\sigma$-finite measure: it is a measure, and no countable cover by finite-measure sets exists	Measure Theory and Integration
a finitely additive set function	a measure: it is additive over finite unions and not countably additive	Measure Theory and Integration
an outer measure on the whole power set	a measure: it is monotone and countably subadditive and additive only on the Carathéodory-measurable sets	Measure Theory and Integration
the set of rationals in $[0,1]$	a set of positive measure: it is countable, hence null, and dense; a null set can be dense	Measure Theory and Integration

Non-Examples of Convergence

Non-example	The positive object it fails to be, and the failure	Introduced in
$f_n = n\mathbf 1_{(0,1/n)}$	an $L^1$-convergent sequence: it converges to $0$ almost everywhere and in measure, while $\int f_n = 1$ throughout	Modes of Convergence
the sliding dyadic intervals on $[0,1]$	an almost-everywhere convergent sequence: it converges to $0$ in measure and in every $L^p$, and at no point of $[0,1)$	Modes of Convergence
$f_n = x^n$ on $[0,1)$	a uniformly convergent sequence: it converges locally uniformly and not uniformly, the rate depending on the point	Modes of Convergence
the unit vectors $e_n$ of $\ell^2$	a norm-convergent sequence: it converges weakly to $0$ and stays at norm $1$	Modes of Convergence
a bounded pointwise convergent sequence of Riemann integrable functions with a non-integrable limit	a sequence for which limit and Riemann integral interchange: the interchange fails without uniform convergence	Real Integration
a pointwise convergent sequence of continuous functions with a discontinuous limit	a uniformly convergent sequence: pointwise convergence does not preserve continuity, uniform convergence does	Modes of Convergence

Non-Examples of Differentiation and Analyticity

Non-example	The positive object it fails to be, and the failure	Introduced in
a differentiable function with a discontinuous derivative	a $C^1$ function: differentiability does not imply the continuity of the derivative	Differential Calculus on Normed Spaces
the function $\lvert x\rvert$	a differentiable function at $0$: continuous everywhere and not differentiable at one point	Real Analysis
a continuous function with a divergent Fourier series	a function whose Fourier series converges: continuity does not force pointwise convergence of the series	Real Harmonic Analysis
a real-analytic function at its radius of convergence	an entire function: the radius of convergence is finite, so the power-series representation, and with it the region of analyticity, is confined to the disc	Analytic Functions and Power Series
the gamma function $\Gamma$	a function with zeros: it has no zeros, only the poles at $0, -1, -2, \dots$, since $1/\Gamma$ is entire	Complex Special Functions
a domain of holomorphy	a domain extended across its boundary: it is maximal, there is a holomorphic function on it that does not extend past the boundary, and the plurisubharmonic exhaustion measures that failure	Several Complex Variables

Non-Examples of Series and Integrals

Non-example	The positive object it fails to be, and the failure	Introduced in
the harmonic series $\sum 1/n$	a convergent series: its terms tend to $0$ and its partial sums diverge	Real Analysis
the alternating harmonic series	an absolutely convergent series: it converges conditionally, and the Archimedean rearrangement paradoxes apply to it	Analytic Functions and Power Series
the integral of $x \mapsto 1/x$ over $[1,\infty)$	a convergent improper integral: the $p$-integral converges exactly for $p > 1$, so this one diverges	Real Integration
the integral of $x \mapsto \sin x/x$ over $(0,\infty)$	an absolutely convergent integral: it converges conditionally and not absolutely	Real Integration
a Dirichlet series at its abscissa of convergence	a series continued holomorphically past the abscissa: by Landau's theorem a series with nonnegative coefficients has a singularity at its abscissa and no continuation across it	Zeta Functions
an infinite sum differentiated term by term without uniform convergence	a differentiable sum with the termwise derivative: the interchange fails without the uniform convergence of the derivatives	Modes of Convergence

Summary

This list gathers the non-examples of analysis that the corpus records: the continuous nowhere differentiable function, the path of Brownian motion, Volterra's function, the Dirichlet function, a non-measurable set and the smooth function that is not analytic among the standard counterexamples; the Cantor function and the singular Cantor law, the fat Cantor set indicator and the non-$\sigma$-finite counting measure among those of measure theory; the sequences that separate the modes of convergence; the differentiable and continuous functions that fail to be smooth or to have convergent Fourier series; and the series and integrals that fail to converge absolutely or at all. Each row names the positive object the non-example fails to be, so that the failure is the content of the row.

Summary of Notation

The non-examples are named rather than denoted; the symbols in the tables are collected here.

Symbol	Meaning
$\mathbf 1_{\mathbb Q}$, $\mathbf 1_A$	the indicator of the rationals, of a set $A$
$\exp(-1/x^2)$	the smooth non-analytic function
$V$	the Vitali non-measurable set
$\ell^2$, $e_n$	the sequence space and its unit vectors
$L^p$, a.e.	the Lebesgue space and the almost-everywhere mode