It's been a long time since I did anything like this, but I'll give it a try. I'm confident in the basic approach of the proof, but I feel like there might be something missing between the first and second step...
Let be a limit point of gr(f).
because f(x) is continuous.
gr(f) is a closed subset of
Suppose that that implies .
Let . From continuity we get
As a sort of lemma, any point in the ball centered at (a,b), .
Can you show that? Hint: .
So suppose some point of the graph is such that .
Can you explain the following contradiction?
That shows that the complement of the is open.