If f {\displaystyle {}f} is not constant, then there exists some x < x ′ {\displaystyle {}x<x'} such that f ( x ) ≠ f ( x ′ ) {\displaystyle {}f(x)\neq f(x')} . Then there exists, due to the mean value theorem, some c {\displaystyle {}c} , x < c < x ′ {\displaystyle {}x<c<x'} , such that f ′ ( c ) = f ( x ′ ) − f ( x ) x ′ − x ≠ 0 {\displaystyle {}f'(c)={\frac {f(x')-f(x)}{x'-x}}\neq 0} , which contradicts the assumption.
