Suppose fi is continuous from a topological space X to the real numbers,how to prove that g=sup{fi:i=1...n} is also continuous?
It is quite clear to show that a function is continuous we must only show that it's continuous with respect to some open subbase for the codomain. So, in particular since if and then forms an open subbase for we must only check the cases for elements of this set.
If then and so and since this is a union of open sets (since each is continuous) it follows that is continuous. The case for is analogous.