Why doesn't Dirichlet function have a derivative in X=0

Question

$\newcommand{\dirichlet}{\mathop{\rm dirichlet}\nolimits}$ I'm trying to find two examples for the following criterias:

1. A method that is continuous in exactly one point but doesn't have a derivative in that point
2. A method that is continuous in exactly one point and does have a derivative in that point

After looking deeper at some examples, I found out that $f(x) = x\cdot\dirichlet(x)$ doesn't have a deriviate at $x = 0$ however $f(x) = x^2\cdot \dirichlet(x)$ does have.

I can't understand the difference between the two. Any helps could help.

Answer 1

First, $\Delta(x)$ is nowhere continuous, e.g. $\lim_{x\to y}\Delta(x)$ does not exist for any $y$. The definition of the derivative is given by a limit, so if you are looking for the derivative of $x^p\Delta(x)$ for $p\geq 1$ at zero, by the definition you have $$ \lim_{x\to 0}\frac{x^p\Delta(x) - 0^p\Delta(0)}{x-0} = \lim_{x\to0}x^{p-1}\Delta(x) $$ and the question now is, when the latter limit does exist. As we discussed already, it does not exist for $p-1=0$ i.e. $p=1$ hence $x\Delta(x)$ does not have a derivative at zero. However, since $$ \lim_{x\to 0}x^{p-1} = 0 $$ for $p>1$ and $\Delta(x)$ is bounded, the limit $\lim_{x\to0}x^{p-1}\Delta(x)$ does exist and equals zero for all $p>1$.