This question already has an answer here:
if $f$ is differentiable at a point $x$, is $f$ also necessary Lipshitz at $x$?
Since $f$ is differentiable at $x$, $f$ is also continuous at $x$. Then we have the $\varepsilon$-$\delta$ definition of $f$ continuous at $x$, and also we have $f'(x)$ exists. But i have no idea how to connect them together. Since $f$ is only differentiable at a point, i don't think mean value theorem is gonna work here either.