Tag Archives: real-analysis

A discontinuous real convex function

September 13, 2015 Jean-Pierre Merx Leave a comment

Consider a function $f$ defined on a real interval $I \subset \mathbb R$ . $f$ is called convex if: $\forall x, y \in I \ \forall \lambda \in [0,1]: \ f((1-\lambda)x+\lambda y) \le (1-\lambda) f(x) + \lambda f(y)$

Suppose that $I$ is a closed interval: $I=[a,b]$ with $a < b$ . For $a < s < t < u < b$ one can prove that: $\frac{f(t)-f(s)}{t-s}\le \frac{f(u)-f(s)}{u-s}\le\frac{f(u)-f(t)}{u-t}.$ It follows from those relations that $f$ has left-hand and right-hand derivatives at each point of the interior of $I$ . And therefore that $f$ is continuous at each point of the interior of $I$ .
Is a convex function defined on an interval $I$ continuous at all points of the interval? That might not be the case and a simple example is the function: $\begin{array}{l|rcl} f : & [0,1] & \longrightarrow & \mathbb R \\ & x & \longmapsto & 0 \text{ for } x \in (0,1) \\ & x & \longmapsto & 1 \text{ else}\end{array}$

It can be easily verified that $f$ is convex. However, $f$ is not continuous at $0$ and $1$ .

Analysis

Counterexamples around differentiation of sequences of functions

June 21, 2015 Jean-Pierre Merx Leave a comment

We consider here sequences of real functions defined on a closed interval. Following theorem is the main one regarding the differentiation of the limit.

Theorem: Suppose $(f_n)$ is a sequence of functions, differentiable on $[a,b]$ and such that $(f_n(x_0))$ converges for some point $x_0 \in [a,b]$ . If $(f_n^\prime)$ converges uniformly on $[a,b]$ , then $(f_n)$ converges uniformly on $[a,b]$ to a function $f$ and for all $x \in [a,b]$ $f^\prime(x)=\lim\limits_{n \to \infty} f_n^\prime(x)$ What happens if we drop some hypothesis of the theorem? Continue reading Counterexamples around differentiation of sequences of functions →

Analysis

Pointwise convergence and properties of the limit (part 1)

May 31, 2015 Jean-Pierre Merx Leave a comment

We look here at the continuity of a sequence of functions that converges pointwise and give some counterexamples of what happens versus uniform convergence.

Recalling the definition of pointwise convergence

We consider here real functions defined on a closed interval $[a,b]$ . A sequence of functions $(f_n)$ defined on $[a,b]$ converges pointwise to the function $f$ if and only if for all $x \in [a,b]$ $\displaystyle \lim\limits_{n \to +\infty} f_n(x) = f(x)$ . Pointwise convergence is weaker than uniform convergence.

Pointwise convergence does not, in general, preserve continuity

Suppose that $f_n \ : \ [0,1] \to \mathbb{R}$ is defined by $f_n(x)=x^n$ . For $0 \le x <1$ then $\displaystyle \lim\limits_{n \to +\infty} x^n = 0$ , while if $x = 1$ then $\displaystyle \lim\limits_{n \to +\infty} x^n = 1$ . Hence the sequence $f_n$ converges to the function equal to $0$ for $0 \le x < 1$ and to $1$ for $x=1$ . Although each $f_n$ is a continuous function of $[0,1]$ , their pointwise limit is not. $f$ is discontinuous at $1$ . We notice that $(f_n)$ doesn't converge uniformly to $f$ as for all $n \in \mathbb{N}$ , $\displaystyle \sup\limits_{x \in [0,1]} \vert f_n(x) - f(x) \vert = 1$ . That's reassuring as uniform convergence of a sequence of continuous functions implies that the limit is continuous! Continue reading Pointwise convergence and properties of the limit (part 1) →

Analysis

A nowhere locally bounded function

May 15, 2015 Jean-Pierre Merx 1 Comment

In that article, I described some properties of Thomae’s function $f$ . Namely:

The function is discontinuous on $\mathbb{Q}$ .
Continuous on $\mathbb{R} \setminus \mathbb{Q}$ .
Its right-sided and left-sided limits vanish at all points.

Let’s modify $f$ to get function $g$ defined as follow:
$g: \left|\begin{array}{lrl} \mathbb{R} & \longrightarrow & \mathbb{R} \\ x & \longmapsto & 0 \text{ if } x \in \mathbb{R} \setminus \mathbb{Q} \\ \frac{p}{q} & \longmapsto & q \text{ if } \frac{p}{q} \text{ in lowest terms and } q > 0 \end{array}\right.$ $f$ and $g$ both vanish on the set of irrational numbers, while on the set of rational numbers, $g$ is equal to the reciprocal of $f$ . We now consider an open subset $O \subset \mathbb{R}$ and $x \in O$ . As $f$ right-sided and left-sided limits vanish at all points, we have $\lim\limits_{n \to +\infty} f(x_n) = 0$ for all sequence $(x_n)$ of rational numbers converging to $x$ (and such a sequence exists as the rational numbers are everywhere dense in the reals). Hence $\lim\limits_{n \to +\infty} g(x_n) = + \infty$ as $f$ is positive.

We can conclude that $g$ is nowhere locally bounded. The picture of the article is a plot of function $g$ on the rational numbers $r = \frac{p}{q}$ in lowest terms for $0 < r < 1$ and $q \le 50$ .

Analysis

A function continuous at all irrationals and discontinuous at all rationals

May 10, 2015 Jean-Pierre Merx 2 Comments

Let’s discover the beauties of Thomae’s function also named the popcorn function, the raindrop function or the modified Dirichlet function.

Thomae’s function is a real-valued function defined as:
$f: \left|\begin{array}{lrl} \mathbb{R} & \longrightarrow & \mathbb{R} \\ x & \longmapsto & 0 \text{ if } x \in \mathbb{R} \setminus \mathbb{Q} \\ \frac{p}{q} & \longmapsto & \frac{1}{q} \text{ if } \frac{p}{q} \text{ in lowest terms and } q > 0 \end{array}\right.$

$f$ is periodic with period $1$

This is easy to prove as for $x \in \mathbb{R} \setminus \mathbb{Q}$ we also have $x+1 \in \mathbb{R} \setminus \mathbb{Q}$ and therefore $f(x+1)=f(x)=0$ . While for $y=\frac{p}{q} \in \mathbb{Q}$ in lowest terms, $y+1=\frac{p+q}{q}$ is also in lowest terms, hence $f(y+1)=f(y)=\frac{1}{q}$ . Continue reading A function continuous at all irrationals and discontinuous at all rationals →

Analysis

A decreasing function converging to zero whose derivative diverges (part2)

March 22, 2015 Jean-Pierre Merx Leave a comment

In that article, I gave examples of real valued functions defined on $(0,+\infty)$ that converge to zero and whose derivatives diverge. But those functions were not monotonic. Here I give an example of a decreasing real valued function $g$ converging to zero at $+\infty$ and whose derivative is unbounded.

We first consider the polynomial map:
$P(x)=(1+2x)(1-x)^2=1-3x^2+2x^3$ on the segment $I=[0,1]$ . $P$ derivative equals $P^\prime(x)=-6x(1-x)$ . Therefore $P$ is decreasing on $I$ . Moreover we have $P(0)=1$ , $P(1)=P^\prime(0)=P^\prime(1)=0$ and $P^\prime(1/2)=-3/2$ . Continue reading A decreasing function converging to zero whose derivative diverges (part2) →

Analysis

Differentiable functions converging to zero whose derivatives diverge (part1)

March 1, 2015 Jean-Pierre Merx 1 Comment

In this article, I consider real valued functions $f$ defined on $(0,+\infty)$ that converge to zero, i.e.:
$\lim\limits_{x \to +\infty} f(x) = 0$ If $f$ is differentiable what can be the behavior of its derivative as $x$ approaches $+\infty$ ?

Let’s consider a first example:
$\begin{array}{l|rcl} f_1 : & (0,+\infty) & \longrightarrow & \mathbb{R} \\ & x & \longmapsto & \frac{1}{x} \end{array}$ $f_1$ derivative is $f_1^\prime(x)=-\frac{1}{x^2}$ and we also have $\lim\limits_{x \to +\infty} f_1^\prime(x) = 0$ . Let’s consider more sophisticated cases! Continue reading Differentiable functions converging to zero whose derivatives diverge (part1) →

Analysis

A continuous function not differentiable at the rationals but differentiable elsewhere

November 30, 2014 Jean-Pierre Merx Leave a comment

We build here a continuous function of one real variable whose derivative exists on $\mathbb{R} \setminus \mathbb{Q}$ and doesn’t have a left or right derivative on each point of $\mathbb{Q}$ .

As $\mathbb{Q}$ is (infinitely) countable, we can find a bijection $n \mapsto r_n$ from $\mathbb{N}$ to $\mathbb{Q}$ . We now reuse the function $f$ defined here. Recall $f$ main properties: Continue reading A continuous function not differentiable at the rationals but differentiable elsewhere →

Analysis

A differentiable function except at one point with a bounded derivative

November 23, 2014 Jean-Pierre Merx 1 Comment

We build here a continuous function of one real variable whose derivative exists except at $0$ and is bounded on $\mathbb{R^*}$ .

We start with the even and piecewise linear function $g$ defined on $[0,+\infty)$ with following values:
$g(x)= \left\{ \begin{array}{ll} 0 & \mbox{if } x =0\\ 0 & \mbox{if } x \in \{\frac{k}{4^n};(k,n) \in \{1,2,4\} \times \mathbb{N^*}\}\\ 1 & \mbox{if } x \in \{\frac{3}{4^n};n \in \mathbb{N^*}\}\\ \end{array} \right.$
The picture below gives an idea of the graph of $g$ for positive values. Continue reading A differentiable function except at one point with a bounded derivative →

Analysis

A continuous function which is not of bounded variation

November 15, 2014 Jean-Pierre Merx Leave a comment

Introduction on total variation of functions

Recall that a function of bounded variation, also known as a BV-function, is a real-valued function whose total variation is bounded (finite).

Being more formal, the total variation of a real-valued function $f$ , defined on an interval $[a,b] \subset \mathbb{R}$ is the quantity:
$V_a^b(f) = \sup\limits_{P \in \mathcal{P}} \sum_{i=0}^{n_P-1} \left\vert f(x_{i+1}) – f(x_i) \right\vert$ where the supremum is taken over the set $\mathcal{P}$ of all partitions of the interval considered. Continue reading A continuous function which is not of bounded variation →

Math Counterexamples

Tag Archives: real-analysis

A discontinuous real convex function

Counterexamples around differentiation of sequences of functions

Pointwise convergence and properties of the limit (part 1)

Recalling the definition of pointwise convergence

Pointwise convergence does not, in general, preserve continuity

A nowhere locally bounded function

A function continuous at all irrationals and discontinuous at all rationals

$f$ is periodic with period $1$

A decreasing function converging to zero whose derivative diverges (part2)

Differentiable functions converging to zero whose derivatives diverge (part1)

A continuous function not differentiable at the rationals but differentiable elsewhere

A differentiable function except at one point with a bounded derivative

A continuous function which is not of bounded variation

Introduction on total variation of functions

Mathematical exceptions to the rules or intuition

Recalling the definition of pointwise convergence

Pointwise convergence does not, in general, preserve continuity

ff is periodic with period 11

Introduction on total variation of functions

Mathematical exceptions to the rules or intuition

$f$ is periodic with period $1$