Let be a differentiable function defined on such that Prove that is a constant function.Problem:

Let be some arbitrary subinterval of . By the mean value theorem we know that there exists some such that but since we have by assumption that . Therefore . Since were arbitrary the conclusion follows.Proof:

Remark:In retrospect, all though I think the above is fine it may have been more intuitive to consider any arbitrary subinterval which would then show that for any arbitrary