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**ThePerfectHacker** Let $\displaystyle c\in \mathbb{R}$, then we know $\displaystyle |f(x) - f(c)| \leq (x-c)^2$.

For $\displaystyle x\not = c$ we have $\displaystyle \left| \frac{f(x) - f(c)}{x-c} \right| \leq |x-c|$.

Thus, the limit of the quotient exists and $\displaystyle \lim_{x\to c} \frac{f(x)-f(c)}{x-c} = 0$.

Thus, $\displaystyle f'(c) = 0$ for all $\displaystyle c\in \mathbb{R}$.

The function $\displaystyle f$ has vanishing derivative everywhere which means we must conclude that $\displaystyle f$ is a konstant function.