integral of a continuous function bounded. Prove function is identically zero. 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 $0$. 

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.
