I'm looking for a function $f$ that is continuous in $x$, i.e. for which holds
$$\lim_{y\to 0} f(x+y)-f(x) = 0$$
, but for which its derivative
$$ \lim_{y\to 0} \frac {f(x+y)-f(x)}y = \infty$$
tends to infinity.
The whole idea of being continuous in a point, yet having an infinite gradient in this point seems like a contradiction, but as differentiable$\implies$continuous is only an implication, it should be possible.
As a possible candidate I've e.g. looked for solutions for the functional equation $f(x+y) = f(x) + g(y)$, where $f$ is continuous,
but as $f(x) = f(0+x) = g(x) $, the only solution for $f(x)$ is $c\cdot x$, which doesn't fulfill the requirement.
Classic example (not the simplest possible, of course):
$$f(x)=\sqrt[3]{x^3-3x^2}$$
Infinite derivatives at $x=0$ and $x=3$, continuous everywhere. I still remember this function from my math exam, like 3 decades ago :)