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 →

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 →

**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 →

## 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 →

## 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 →

## Mathematical exceptions to the rules or intuition