1. The problem statement, all variables and given/known data Let F: X x Y -> Z. We say F is continuous in each variable separately if for each ##b \in Y## the function h: X -> Z, h(x) = F(x,b), and for each ##a \in X##, the function g: Y -> Z, g(y) = F(a,y) is continuous. Show that if F is continuous, then F is continuous in each variable separately. 2. Relevant equations 3. The attempt at a solution So I noticed that you can define a function A: X -> X x Y and B: Y -> X x Y, and h = A ° F and g = B ° F. If I can show that A and B are continuous then I am done. However I do not see how to do that. I have a theorem in my book that says, Let the Topo space M be a subspace of the topo space N. Then the inclusion mapping i: M -> N is continuous. Can I use this for A and B? Can I just say they are inclusion mappings?