Problem Statement: is continuous on . Given that
(a) prove has continuous derivative on (Hint: start with the change of variable u=x+t). (b) Given and show that there exists a such that for every .
(a)Proof: First let us start by making the change of variable suggested in the problem statement.
Let , then
Now we may apply the Fundamental Theorem of Calculus and obtain that
which is continuous since is continuous on
(b)Proof: We are given and . Since is continuous on a compact set it is uniformly continuous, thus, for our given we may find a such that whenever . Now consider the following:
And so, for it follows that . This allows us to simplify further:
Reflection: Always try the easiest method first!!! This is a lesson I keep “learning” but I don’t really seem to be learning it. Part (b) I attempted to do in a very strange way. Hopefully now I have really learned, try the easier method first!! If it doesn’t work then at least you’ll have gained some insight into why it isn’t working, and that can help you with another attempt.