A Function with Small Derivative Has Exactly One Fixed Point

In the setting of Single-Variable Calculus on an Interval, in which $\mathbb{R}$ is recorded as an interval every point of which is an interior point of it, let $f:\mathbb{R}\to\mathbb{R}$ be differentiable at every point of $\mathbb{R}$, and suppose that

$$|f'(x)|\le\tfrac12\qquad\text{for every }x\in\mathbb{R}.$$

Problem. Show that there is exactly one real number $c$ with $f(c)=c$.