Topology sucks my ass. >:(

If X,Y,Z are topological spaces, and f: X x Y -> Z is a continuous map, then for a fixed y in y, is g(x) := f(x,y) necessarily continuous as a function from X -> Z? 

It better be, or else I'm even more confused than I thought.  Oh, and X, Y, and Z are Hausdorff, but I don't know if it matters.

And no, it's not homework, so stfu about that. :p

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f is continuous and i is continuous by the definition of the product topology so g is continuous.*

I'm a retard today. ;_;