Suppose are both continuous on D. Define by . Show that h is continuous on D.
Case 1:
Case 2:
Are these the two cases, and if so, I am not sure how to prove this.
Yes, but we still don't cover all the cases (the example I gave shows that we can have neither the first case (and the similar) nor the second.
You can write and you have to use the fact that the sum of two continuous functions is continuous, and if is continuous then so is .