Limit of a continuous function Suppose that $f$ is a continuous and real function on $[0,\infty]$. How can we show that if $\lim_{n\rightarrow\infty}(f(na))=0$ for all $a>0$ then $\lim_{x\rightarrow+\infty} f(x)=0$? 

This is a standard (moderately tough) exercise in applying the Baire category theorem. Tim Gowers did a presentation of this result on his blog under the title "What is deep mathematics?".
But as he writes: 

If you haven’t seen this before and want to get the most out of this post then you should (of course) make a serious attempt to solve this beautiful problem before reading on.

