Expii

Function with Always-Zero Derivative Is Constant - Expii

Suppose a function f(x) is continuous on [a,b] and differentiable on (a,b). If f is constant, then of course it has always-zero derivative. Conversely, if f'(x)=0 on (a,b) (in other words, if the derivative vanishes everywhere on (a,b)), then f must be constant. This observation will come in handy when we discuss anti-derivatives later on.