construct a function f defined in the compact interval [0,1] of the real line that for some x irrational the limit

Question

construct a function f defined in the compact interval $[0,1]$ of the real line that for some x irrational the limit:

$\lim\limits_{N \to \infty}\frac{1}{N}\sum\limits_{k=1}^{N} F(kx \bmod 1)\neq\int\limits_0^1F(t)dt$

Answer 1

Hint: there are only countably many values $kx \mod 1$.