# An unbounded positive continuous function with finite integral

Consider the piecewise linear function $$f$$ defined on $$[0,+\infty)$$ taking following values for all $$n \in \mathbb{N^*}$$:
$f(x)= \left\{ \begin{array}{ll} 0 & \mbox{if } x=0\\ 0 & \mbox{if } x=n-\frac{1}{2n^3}\\ n & \mbox{if } x=n\\ 0 & \mbox{if } x=n+\frac{1}{2n^3}\\ \end{array} \right.$

The graph of $$f$$ can be visualized in the featured image of the post. Continue reading An unbounded positive continuous function with finite integral

# A continuous differential equation with no solution

Most of Cauchy existence theorems for a differential equation

\textbf{x}^\prime = \textbf{f}(t,\textbf{x})
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

# A nowhere continuous function

This is a strange function!

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.