On the domain , is continuous?

If we let , then it is clear that for all , is continuous, so now it will suffice to show that uniformly converges to for . So for any , there exists such that if , then .

This is where I am running into some trouble because . After this point I am not exactly sure where to proceed. I have a hunch that the fact that converges on that domain will come into play, but I am not sure. Thanks.