Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is differentiable and $\left| f ^ { \prime } ( x ) \right| < \frac { 1 } { 2 }$ for all $x \in \mathbb { R }$. Show that for some $x _ { 0 } \in \mathbb { R } , f \left( x _ { 0 } \right) = x _ { 0 }$.