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Let $f$ be a continuous map $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be such that $$f(x,y+L)=f(x,y),\quad L\in \mathbb{R}, L>0$$ and $$\frac{\partial f}{\partial x}\neq 0$$

Suppose $$F(x,z)=\displaystyle\int_{z}^{z+L} f(x,y) dy$$ and that $$\frac{\partial F}{\partial x}= 0=\frac{\partial F}{\partial z},$$ i.e. $F$ is constant. Is $F$ identically zero?

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    Consider $f(x,y) = 1 + g(x)\cos(2\pi y/L)$, for some arbitrary function $g$. – 2017-01-18
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    @eyeballfrog Interesting, thank you! – 2017-01-18

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