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

who the hell cares

Yes it is continuous.

A restriction of a continuous function is still continuous, since keeping within delta of (x,y) still keeps you within epsilon of (f(x),f(y)).

Err, sorry, accidentally phrased that in terms of metric spaces. Meant to say:

g(x) = f(i(x)) where i(x) = (x,y).

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

WLDLWMDLKMWD£$~"$£$%£%~"@$£

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

I'm a retard today. ;_;

>>5
FFUUUUUUU, yeah, that works.  I'm such a tard lol. >_>

You don't want to know how long I spent staring blankly at that last night.  Thanks so much.