I am reading "Dialogues on Calculus" (in Japanese) by Kohji Kasahara.
Hotta is one of the fictional characters in this dialogues.
Hotta says as follows:
If $f$ is analytic at $a$, it is analytic at any point in a neighborhood of $a$.
This is unbelievable.
If $f$ is continuous at $a$, it it possible $f$ is not continous at some point in any neighborhood of $a$ in general.
If $f$ is differentiable at $a$, it it possible $f$ is not differentiable at some point in any neighborhood of $a$ in general.
I found $f$ which is continuous at $a$, but it is not continuous at some point in any neighborhood of $a$:
Let $f$ be a function such that $f(0)=0$ and $f(x)=x$ for any $x\in\mathbb{Q}\setminus\{0\}$ and $f(x)=0$ for any $x\in(\mathbb{R}\setminus\mathbb{Q})$.
I was not able to find $f$ which is differentiable at $a$, but it is not differentiable at some point in any neighborhood of $a$.
Please tell me an example.