 Solution

Suppose distinct fixed points $x<y$ existed. Then $h(t)=f(t)-t$ satisfies $h(x)=h(y)=0$. Rolle's theorem supplies $c\in (x,y)$ with $h^{\prime }(c)=0$, hence $f^{\prime }(c)=1$, contrary to the hypothesis. Thus there is at most one fixed point. The argument does not require $f^{\prime }$ to be continuous.