Back studying some math and I am not sure the argument below is rigorous.
Let $f(.)$ be a continuously differentiable function taking values on a compact interval. By the mean value theorem:
$$ \exists t, f(a) - f(b) = f'(t)(a-b) $$
By continuity and differentiability of $f$, then can I conclude below?
$$ f(a) - f(b) \geq f'(t+\varepsilon)(a-b) ~ or ~ f(a) - f(b) \geq f'(t-\varepsilon)(a-b) $$
-Is this legitimate? -Do I need a continuous derivative or only continuity and differentiability suffices?