Is there a function that has continuity in its first variable but not in its second? Defining continuity in a variable like this: $\phi(x)= f(x,y_0)$ continuity in $x_0$, and $\Upsilon(y)= f(x_0,y)$ is continuity in $y_0$.
I can't understand. If you can put examples, it would be great.
How about $f(x,y)=g(x)+h(y)$, where g is continuous and h is discontinuous?. So for instance $f(x,y)=x-\dfrac1{y-1},y \le 0$ and $f(x,y)=x-\dfrac1{y+1}, y\gt 0$ would be continuous in x but discontinuous in y at (0,0)