Suppose $f$ is differentiable on $[0,\infty)$. One may write \frac{f(t)-f(0)}{t-0} = f'(\epsilon) for $t\in (0,1),\epsilon \in (0,t)$ by the Mean Value Theorem. I would like to then take the limit as $t\to 0$ (so $\epsilon \to 0$ as well) and say that this equals f'(0). But I think this would require continuity of the derivative, because we don't know how $\epsilon$ changes.
However, if we just apply the definition of $f$ being differentiable we get \lim_{t\to 0}\frac{f(t)-f(0)}{t-0} = f'(0) by definition.
This doesn't feel right to me. I wouldn't think that applying the Mean Value Theorem would lose so much information that I cannot justify the limit anymore.
I there a way to show the limit exists using the Mean Value Theorem like I tried? (Of course using the definition is better, but I just want to know if there is a way to do it with MVT).