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**kevinlightman** Prove that if g:R->R is continuous at a then f(x,y)=g(x) is continuous at (a,b) $\displaystyle \forall b \in R$

So we know

$\displaystyle \forall e>0 \exists d>0 s.t. \forall x \in R$ where |x-a|<d we have |g(x) - g(a)|<e

So I've said as$\displaystyle \forall b \in R$ 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 $\displaystyle [(x-a)^2 + (y-b)^2]^\frac{1}{2}<d$

This seems to be an incorrect cheat though, am I along the right lines or not?