$f(x)=f(x+T),T\notin \mathbb Q,\int_0^T f(x)\mathrm dx=-T$. Does ther exist a function $f$ such that $\sup \sum_{k_1

Suppose $f\in C(\mathbb R )$, and we take supremum from all $x\in\mathbb R,k_1,k_2\in\mathbb Z$.

I doubt that there won't exist such a function, for $k_1,k_2$ are integers but the function's period is irrational. But I can't clearly turn this thought into formal proof. Any help would be appreciated!