Theorem: Let and be metric spaces. A function is continuous if and only if for every open , then
is open in .
If I want to show that the function defined by
is continuous, then I need to consider any open set and show that
I understand that if and , then . But I don't understand what happens when only , or only , or .
If only , then is it correct to say that =[0,1]? If only , then is it correct to say that =[2,4]? But I am really confused since these are closed intervals. I am lost.
Can someone help me to find for those cases? Any clues or hints will be appreciated.