Let f be a differentiable real function and let M be a positive real number. Prove that if |f(x\plus{}t)\minus{}2f(x)\plus{}f(x\minus{}t)| \leq Mt^2 \; \textrm{for all}\ \;x\ \; \textrm{and}\ \;t\ , then |f'(x\plus{}t)\minus{}f'(x)| \leq M|t|.
J. Szabados functionreal analysisreal analysis unsolved