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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

8 votes
3 answers
823 views

Second order difference implies differentiability

Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies $|f'(x+h)-f'(x)|\le c |h| …