The existence of a strange function on $[0,1]^2$

Question

Does such a function $f:[0,1]^2\rightarrow \mathbb{R}$ exist:

1. $f(\cdot,y)$ and $f(x,\cdot)$ are continuous functions with respect to "$\cdot$", for any $x,y\in[0,1]$;

2. The zero set of $f$ is dense in $[0,1]^2$;

3. $f$ is not identically zero.

Any help, any hint, any reference will be appreciated.