Bounded derivative implies uniform continuity

Bounded derivative implies uniform continuity: Let $I\subseteq\mathbb R$ be an interval and let $f:I\to\mathbb R$ be differentiable on $I$. If there is a constant $M$ such that $|f'(t)|\le M$ for all $t\in I$, then for all $x,y\in I$,

Consequently, $f$ is Lipschitz and in particular uniformly continuous on $I$.

This estimate is a direct application of the mean value theorem . It is the one-dimensional special case of the mean value inequality for differentiable maps.