For a short reminder about **ordered fields** you can have a look to following post. We prove there that \(\mathbb{Q}\) can be ordered in only one way.

That is also the case of \(\mathbb{R}\) as \(\mathbb{R}\) is a **real-closed field**. And one can prove that the only possible **positive cone** of a real-closed field is the subset of squares.

However \(\mathbb{Q}(\sqrt{2})\) is a subfield of \(\mathbb{R}\) that can be ordered in two distinct ways. Continue reading A field that can be ordered in two distinct ways →

## Introduction to ordered fields

Let \(K\) be a field. An **ordering** of \(K\) is a subset \(P\) of \(K\) having the following properties:

- ORD 1
- Given \(x \in K\), we have either \(x \in P\), or \(x=0\), or \(-x \in P\), and these three possibilities are mutually exclusive. In other words, \(K\) is the disjoint union of \(P\), \(\{0\}\), and \(-P\).
- ORD 2
- If \(x, y \in P\), then \(x+y\) and \(xy \in P\).

We shall also say that \(K\) is **ordered by \(P\)**, and we call \(P\) the set of **positive elements**. Continue reading An infinite field that cannot be ordered →

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:

\(f_1\) is continuous everywhere and differentiable on \(\mathbb{R} \setminus \mathbb{Z}\). Continue reading A function that is everywhere continuous and nowhere differentiable →

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 \(C\).

## The specific case of dimension \(2\)

**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 →

## Mathematical exceptions to the rules or intuition