Zero derivative implies locally constant
For an interval in the domain
Suppose is a function and is an interval (possibly open, closed, or half-open and half-closed) contained inside the domain of such that for all in the interior of and is continuous on all of (note that we do not require differentiability, or even one-sided differentiability, at the endpoints of , if any). Then, is a constant function on all of , i.e. there is a real number such that for all .