If $g(x):=f(x, kx^m)$ is continuous at $0\;\;\;$ $\forall k\in R$, $\;\;\;\forall m\in N$, then $f(x,y)$ is continuous at $(0,0)$.
I'm not quite sure what is meant here by at 0. This means $f(x, kx^m) = 0$? Seems there could be an issue in this statement when k = 0?