# Function constant a.e.

Suppose that $f:[0,1]^2\longrightarrow{\mathbb{R}}$ is a function such that $f(.,y)$ is constant for almost all $y$, and $f(x,.)$ is constant for almost every $x$. Prove that $f$ is constant ctp (with respect to u, where u is the Lebesgue measure).
For each y fixed $f(.,y):[0,1] \times \left\{ y \right\}\longrightarrow{\mathbb{R}}$