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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?

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  • $\begingroup$ If $f'$ is increasing or decreasing. $\endgroup$ – hamam_Abdallah Oct 6 '18 at 19:56

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