I'm a little confused with the definition of a function in metric spaces.
Suppose that $(M,d_1)$, $(N,d_2)$ and $(P,d_2)$ are metric spaces, and $f:M \times N\rightarrow P$ is a function of $M\times N$ into N, and $(a,b) \in M$. Then, $f$ is continuous on $(a,b)$ if, for every $ \delta_1,\delta_2>0$, $x \in M$ and $y \in N$ $d_1(x,a)<\delta_1$, $d_2(y,b)<\delta_2$ then, $d(f(x,y),f(a,b))<\epsilon$.
This is the correct definition of continuous function of two variable function? Works for every metric or is specific for the max distance metric? And how could I define separeted continuity?