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Let ${\rm f}:{\mathbb R}^{k} \to {\mathbb R}$ be a continuous function. Assume that for any $a > 0$ and any $k$-cell $Q_{a}$ of side length $a$ $\left(~\mbox{and therefore volume}\ a^{k}\right)$ we have

$\displaystyle{% \left\vert\,\int_Q {\rm f}\left(x\right)\,{\rm d}x\,\right\vert\le a^{2k} }$

Prove that $f$ is identically $0$.

Felix Marin
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user2496006
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1 Answers1

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Use the fact that if $Q_a$ is centered at $x_0$ then $$\lim_{a \to 0^+} \frac{1}{a^k} \int_{Q_a} f(x) \, dx = f(x_0).$$ Your hypothesis implies that this limit is zero.

Umberto P.
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