Zero derivative implies locally constant

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For an interval in the domain

Suppose f is a function and I is an interval (possibly open, closed, or half-open and half-closed) contained inside the domain of fsuch that f'(x) = 0 for all x in the interior of I and f is continuous on all of I (note that we do not require differentiability, or even one-sided differentiability, at the endpoints of I, if any). Then, f is a constant function on all of I, i.e. there is a real number C such that f(x) = C for all x \in I.

Facts used

  1. Lagrange mean value theorem