g is a Lipschitz on the interval [α-A,α+A] for some A>0. Prove |g'(α)|≤λ

Question

Suppose that g is a C1 function such that the Lipschitz estimate |g(x)-g(y)|≤λ|x-y| holds an interval [α-A,α+A] for some A>0. Prove that |g'(α)|≤λ. (Hint: consider the difference quotients used to define g'(α).)

Answer 1

hint: $$g'(\alpha) = \lim_{h \to 0} \frac{g(\alpha + h) - g(\alpha)}{h}$$ Can we manipulate this into something more useful?