Another necessary condition for a continuous $f$ would be: $f$ should not have a unique global maximum or global minimum at any $0<a<1$ over $[0,t]$ where $t=\arg\max_{0\leq x\leq 1}(x + \sin(\pi x)]$. Also, let $g(x) = f(x+\sin(\pi x))$. 

Suppose it does have a global maximum over $[0,t]$ at some $0<a<1$. Let $b$ be the smallest positive real such that $b+\sin(\pi b)=a$. Certainly, $0<b<a$. Now, $g(b)=f(a)>f(b)$, since $f(a)$ is the maximum. Similarly, $g(a)<f(a)$. This implies that $f$ and $g$ must intersect somewhere within $[b,a]$, a contradiction. A similar argument hold is there existed a global minimum.

This condition can be applied to $[2m,2m+t]$, for any integer $m$. Hope this helps.