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→
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→
Let \(f_1(x) = |x|\) for \(| x | \le \frac{1}{2}\), and let \(f_1\) be defined for other values of \(x\) by periodic continuation with period \(1\). \(f_1\) graph looks like following picture:
Consider the piecewise linear function \(f\) defined on \([0,+\infty)\) taking following values for all \(n \in \mathbb{N^*}\):
\[
f(x)=
\left\{
\begin{array}{ll}
0 & \mbox{if } x=0\\
0 & \mbox{if } x=n-\frac{1}{2n^3}\\
n & \mbox{if } x=n\\
0 & \mbox{if } x=n+\frac{1}{2n^3}\\
\end{array}
\right.
\]
Most of Cauchy existence theorems for a differential equation
\begin{equation}
\textbf{x}^\prime = \textbf{f}(t,\textbf{x})
\end{equation} where \(t\) is a real variable and \(\textbf{x}\) a vector on a real vectorial space \(E\) are valid when \(E\) is of finite dimension or a Banach space. This is however not true for the Peano existence theorem. Continue reading A continuous differential equation with no solution→
One example is the Dirichlet function \(1_{\mathbb{Q}}\).
\(1_{\mathbb{Q}}(x)=1\) if \(x \in \mathbb{Q}\) and
\(1_{\mathbb{Q}}(x)=0\) if \(x \in \mathbb{R} \setminus \mathbb{Q}\).
\(1_{\mathbb{Q}}\) is everywhere discontinuous because \(\mathbb{Q}\) is everywhere dense in \(\mathbb{R}\).
The function \(x \mapsto x \cdot 1_{\mathbb{Q}}(x)\) is continuous in \(0\) and discontinuous elsewhere.
