A decreasing function converging to zero whose derivative diverges (part2)

In that article, I gave examples of real valued functions defined on $$(0,+\infty)$$ that converge to zero and whose derivatives diverge. But those functions were not monotonic. Here I give an example of a decreasing real valued function $$g$$ converging to zero at $$+\infty$$ and whose derivative is unbounded.

We first consider the polynomial map:
$P(x)=(1+2x)(1-x)^2=1-3x^2+2x^3$ on the segment $$I=[0,1]$$. $$P$$ derivative equals $$P^\prime(x)=-6x(1-x)$$. Therefore $$P$$ is decreasing on $$I$$. Moreover we have $$P(0)=1$$, $$P(1)=P^\prime(0)=P^\prime(1)=0$$ and $$P^\prime(1/2)=-3/2$$. Continue reading A decreasing function converging to zero whose derivative diverges (part2)

One matrix having several interesting properties

We consider a vector space $$V$$ of dimension $$2$$ over a field $$\mathbb{K}$$. The matrix:
$A=\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$ has several wonderful properties!

Only zero as eigenvalue, but minimal polynomial of degree $$2$$

Zero is the only eigenvalue. The corresponding characteristic space is $$\mathbb{K} . e_1$$ where $$(e_1,e_2)$$ is the standard basis. The minimal polynomial of $$A$$ is $$\mu_A(X)=X^2$$. Continue reading One matrix having several interesting properties

A separable space that is not second-countable

In topology, a second-countable space (also called a completely separable space) is a topological space having a countable base.

It is well known that a second-countable space is separable. For the proof consider a second-countable space $$X$$ with countable basis $$\mathcal{B}=\{B_n; n \in \mathbb{N}\}$$. We can assume without loss of generality that all the $$B_n$$ are nonempty, as the empty ones can be discarded. Now, for each $$B_n$$, pick any element $$b_n$$. Let $$D=\{b_n;n \in \mathbb{N}\}$$. $$D$$ is countable. We claim that $$D$$ is dense in $$X$$. To see this let $$U$$ be any nonempty open subset of $$X$$. $$U$$ contains some $$B_p$$, hence $$b_p \in U$$. So $$D$$ intersects $$U$$ proving that $$D$$ is dense.

What about the converse? Is a separable space second-countable? The answer is negative and I present below a counterexample. Continue reading A separable space that is not second-countable

Differentiable functions converging to zero whose derivatives diverge (part1)

In this article, I consider real valued functions $$f$$ defined on $$(0,+\infty)$$ that converge to zero, i.e.:
$\lim\limits_{x \to +\infty} f(x) = 0$ If $$f$$ is differentiable what can be the behavior of its derivative as $$x$$ approaches $$+\infty$$?

Let’s consider a first example:
$\begin{array}{l|rcl} f_1 : & (0,+\infty) & \longrightarrow & \mathbb{R} \\ & x & \longmapsto & \frac{1}{x} \end{array}$ $$f_1$$ derivative is $$f_1^\prime(x)=-\frac{1}{x^2}$$ and we also have $$\lim\limits_{x \to +\infty} f_1^\prime(x) = 0$$. Let’s consider more sophisticated cases! Continue reading Differentiable functions converging to zero whose derivatives diverge (part1)