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The statement "A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval." is in Wikipedia. But I don't know how to prove it. In addition, what is difference between non-decreasing and monotonically non-decreasing?

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    There is no difference between non decreasing and monotonically non decreasing. Look at the function $R$ in https://en.wikipedia.org/wiki/Convex_function#Functions_of_one_variable. – copper.hat Oct 22 '22 at 02:32

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what is the difference between non-decreasing and monotonically non-decreasing?

A non-decreasing function is a function that $\forall a < b \ (f(a) \leq f(b))$ while a monotonically non-decreasing function is a function that $\forall a < b \ (f(a) < f(b))$.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval.

The answer can be found on Wikipedia.

Tiago Cavalcante
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  • Why $\lim_{\Delta \to 0} \ f(a + \Delta) - f(a) \leq \lim_{\Delta \to 0} \ f(b + \Delta) - f(b)$ implies $f(a) \leq f(b)$? –  Oct 22 '22 at 15:10
  • @Justin it was wrong in truth, $f(x) = x^2$ has a increasing derivative, but $f(x)$ isn't increasing on every interval. – Tiago Cavalcante Oct 22 '22 at 16:41