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?