If I know a function is continuous on $[0, 1]$, and that $\lim_{h \in \mathbb Q \to 0} {{f(c+h)-f(c)}\over h }$ exists, can I say $f'(c)$ exists?
I feel like this is an obvious yes, since by definition of derivative, $f'(c) = \lim_{h \to 0} {{f(c+h)-f(c)}\over h }$. Hence if that limit exists and is finite, the function at $c$ is differentiable, and so $f(c)$ exists.
Does the fact that $h \in \mathbb Q$ change anything?
And if so, where does $f(x)$ being continuous on $[0,1]$ come in?