g(x, y) = x

I am trying to prove that g is continuous on $\displaystyle R^2$ using the epsilon-delta definition of continuity.

I gave it a try but it doesn't look right:

Given $\displaystyle c \in R$ and $\displaystyle \epsilon > 0$, set $\displaystyle \delta = \epsilon $

Then whenever $\displaystyle \| x-c \| < \delta$, it follows that $\displaystyle \| g(x, y) - g(c, y) \| < \epsilon $

Thanks for any help in advance.