A continuous function not differentiable at the rationals but differentiable elsewhere

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

A differentiable function except at one point with a bounded derivative

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

Converse of Lagrange’s theorem does not hold

Lagrange’s theorem, states that for any finite group \(G\), the order (number of elements) of every subgroup \(H\) of \(G\) divides the order of \(G\) (denoted by \(\vert G \vert\)).

Lagrange’s theorem raises the converse question as to whether every divisor \(d\) of the order of a group is the order of some subgroup. According to Cauchy’s theorem this is true when \(d\) is a prime.

However, this does not hold in general: given a finite group \(G\) and a divisor \(d\) of \(\vert G \vert\), there does not necessarily exist a subgroup of \(G\) with order \(d\). The alternating group \(G = A_4\), which has \(12\) elements has no subgroup of order \(6\). We prove it below. Continue reading Converse of Lagrange’s theorem does not hold

A continuous function which is not of bounded variation

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

A curve filling a square – Lebesgue example

Introduction

We aim at defining a continuous function \(\varphi : [0,1] \rightarrow [0,1]^2\). At first sight this looks quite strange.

Indeed, \(\varphi\) cannot be a bijection. If \(\varphi\) would be bijective, it would also be an homeomorphism as a continuous bijective function from a compact space to a Haussdorff space is an homeomorphism. But an homeomorphism preserves connectedness and \([0,1] \setminus \{1/2\}\) is not connected while \([0,1]^2 \setminus \{\varphi(1/2)\}\) is.

Nor can \(\varphi\) be piecewise continuously differentiable as the Lebesgue measure of \(\varphi([0,1])\) would be equal to \(0\).

\(\varphi\) is defined in two steps using the Cantor space \(K\). Continue reading A curve filling a square – Lebesgue example