Let $f$ be a continuous function on $\mathbb R$ with period $1$. Show that for any irrational number $x$, $\frac { \sum_{k =1 }^ N f (k x ) } { N} \to \int _ 0 ^1 f(t) \, d t $ as $N \to \infty$.
I know that $[kx]$ will be dense in $[0,1]$. But for this problem this seems to be not enough. I feel that I need to know something about how $[kx]$ spreads "evenly" throughout $[0,1]$ as $k$ grows in order to derive the limit. Any help is appreciated.