Can someone check my proof for non-differentiability?

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 haveso that cannot exist. **QED.**

Any help is much appreciated!