Hi all. Doing a problem led me to derive this little lemma, but I'd like someone to check it for me if you would be so kind.
Theorem: Let be an integrable function and define Suppose that, for some in the domain of we have where the limits are presumed to exist. Then is not differentiable at
Proof: Assume without loss of generality that Then there is some such that implies:
- for
- for
Clearly Then for any we haveand for any we have
so that cannot exist. QED.
Any help is much appreciated!