A positive real polynomial function of one variable is always having a minimum.

This is not true for polynomial functions of two variables or more. Continue reading A positive polynomial not reaching its infimum

A positive real polynomial function of one variable is always having a minimum.

This is not true for polynomial functions of two variables or more. Continue reading A positive polynomial not reaching its infimum

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

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

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.