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

Let’s remind that an extreme point \(c\) of a convex set \(C\) in a real vector space \(E\) is a point in \(C\) which does not lie in any open line segment joining two points of \(C\).

**Proposition**: when \(C\) is closed and its dimension is equal to \(2\), the set \(\hat{C}\) of its extreme points is closed.

Continue reading A compact convex set whose extreme points set is not close

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

Look at pi-Base for a gold mine of topological examples.

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

Want to know more about **counterexample** definition? Look at Wikipedia.

We consider a decreasing sequence \((C_n)_{n \in \mathbb{N}}\) of *non empty closed convex subsets of a Banach space* \(E\).

If the convex subsets are closed balls, their intersection is not empty. To see this let \(x_n\) be the center and \(r_n > 0\) the radius of the ball \(C_n\). For \(0 \leq n < m\) we have \(\Vert x_m-x_n\Vert \leq r_n – r_m\) which proves that \((x_n)_{n \in \mathbb{N}}\) is a Cauchy sequence. As the space \(E\) is Banach, \((x_n)_{n \in \mathbb{N}}\) converges to a limit \(x\) and \(x \in \bigcap_{n=0}^{+\infty} C_n\). Continue reading An empty intersection of nested closed convex subsets in a Banach space

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.