Will we have uniform convergence in this case?

Question

Assume that $f: [a,b] \rightarrow \mathbb{R}$ is continuously differentiable.

Then we know that for each $t \in [a,b]$ we have that

$$\frac{f(t+\Delta t)-f(t)}{\Delta t}-f'(t)$$

will converge to zero as $\Delta t\rightarrow 0$. But I am wondering, will this convergence be uniform?

Answer 1

Hint:

By the MVT we have$\left|\frac{f(t + \Delta t) - f(t)}{\Delta t} - f'(t)\right| = |f'(c) - f'(t)|$ where $c$ is between $t$ and $t + \Delta t$. Now apply uniform continuity of $f'$.

Answer 2

As stated, since $f'(t) = \lim\limits_{\Delta t \to 0} \frac{f(t+\Delta t -f(t)}{\Delta t}$ then all you have is 0 which is uniformly continuous. You already stated it's continuously differentiable so these two ways of writing it are the same.