Mean value inequality

If the line segment from $x$ to $y$ lies in the domain of a differentiable map and $\Vert D{f}_z\Vert \le M$ along it, then

A differentiable map on a connected open subset of Euclidean space whose derivative vanishes everywhere is constant. The mean value inequality makes it constant on every ball in the domain, hence locally constant; connectedness then makes the value global.