Suppose the statement is false; no such exists.
Then the function being continuous has absolute minimum of
Call it . Now it should be clear that .
Could we have such that .
This is a problem that my professor has in his notes. I could really use some guidance. I don't understand how to prove problems such as these.
Let be continuous. If there is a , such that for any , then prove that there is a such that . [Hint: Consider g(x)=|f(x)|]