Derivative zero implies constant

If the derivative of a differentiable function is zero everywhere on an interval, the function is constant.

Derivative zero implies constant

Derivative zero implies constant: Let $I\subseteq\mathbb{R}$ be an interval , and let $f:I\to\mathbb{R}$ be differentiable on $I$ . If

f'(x)=0 \quad \text{for all } x\in I,

then $f$ is constant on $I$ .

This is a direct application of the mean value theorem : the derivative controls differences $f(y)-f(x)$ on intervals. It can be viewed as a special case of derivative sign implies monotonicity applied in both directions.