1. The problem statement, all variables and given/known data Find a function f on [-1,1] such that :- (a) there exists c [itex]\in[/itex] (-1,1) such that f'(c) = 0 and (b) f(a) [itex]\neq[/itex] f(b) for any a[itex]\neq[/itex] b [itex]\in[/itex] [-1,1] 2. Relevant equations Lagrange's Mean Value Theorem (LMVT), which states that if f:[a,b]-->ℝ is a function which is continuous in [a,b] and differentiable in (a,b), there exists some c[itex]\in[/itex] (a,b) such that f'(c) = (f(b) - f(a))/(b-a). 3. The attempt at a solution If f was said to be differentiable in (-1,1), then it's quite easy to prove that such a function cannot exist, using LMVT. But here, nothing is said about the continuity or differentiability of f in (-1,1). The function could just be differentiable at the point x = c. I'm quite sure that such a function cannot exist. However, this is purely intuitive, and I'm unable to prove it. Can someone please hint on a proof, or an example if there's any? Thank you.