Constancy theorem ($f^{\prime }=0⟹f\text{ constant on an interval}$)

A differentiable function on an interval whose derivative is identically zero is a constant function. Indeed, for any two arguments $x<y$, the mean value theorem on $[x,y]$ gives $f(y)-f(x)=f^{\prime }(c)(y-x)=0$. For an interval with endpoints included, continuity at those endpoints is required. A disconnected domain may have different constants on different components.