The order of your quantifiers is wrong. In particular, implies uniform continuity, not continuity...
You seem to have the right answer though; you can use the same epsilon and delta.
Prove that if g:R->R is continuous at a then f(x,y)=g(x) is continuous at (a,b)
So we know
where |x-a|<d we have |g(x) - g(a)|<e
So I've said as g(x)=f(x,y) & g(a)=f(a,b), these can be substituted in giving the expression we need except for the condition that
This seems to be an incorrect cheat though, am I along the right lines or not?