Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ and $g : \mathbb { R } \longrightarrow \mathbb { R }$ be continuous functions. Assume that $g$ is periodic with period 1. Show that
$$\lim _ { n \mapsto \infty } \int _ { 0 } ^ { 1 } f ( x ) g ( n x ) d x = \left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right)$$