Problem Statement:Suppose that is such that, for some , if , then . Prove that is constant.
I dont see how this is true. for instance, . Is there something I am missing?
You could state this a little bit more rigorously.
1) Let then the above says that . Thus, since we have that (by continuity of ) and thus by the squeeze theorem is differentiable at and . Since was arbitrary it follows that for all
2) Since is an interval we may conclude that .
Just a comment though.