I have this exercise for my subject Differentials Equations II:
Let $\emptyset\neq I\subset\mathbb{R}$ an interval and $f:\longrightarrow \mathbb{R}$ a function. Let's denote
$I^*=\{t\in I:\exists f'(t)\}$
Suppose that exists a sequence $t_n\in I^*$ such $|f'(t_n)|\longrightarrow\infty$
Prove that $f$ is not lipschitzian in I.
I have tried to prove this by showing that there's a point where $f'$ is not bounded. That point should be the limit of the sequence $t_n$, but I do not know if $t_n$ is convergent. In addition, I do not know if $f$ is continous (I do not know if that's an error from my teacher, because if we are talking about lipschitzian functions it should be at least continous I suppose).
Any hints?