I am trying to find an example, preferably an explicit one, of a differentiable function $g:\mathbb{R}\rightarrow \mathbb{R}$ satisfying the following conditions:
$\displaystyle g(0)=0, g(1)=1, g(-1)=-1;$
$\displaystyle g^\prime(1)=g^\prime(-1)=\frac{1}{2};$
$\displaystyle g^\prime(x)\geq\frac{1}{2} \quad \forall x\in [-1,1].$
The function $g(x)=x$ satisfies the first and the last conditions, but we need to modify it, at least locally around the points $x=\pm1$, to meet the constraints about derivatives at the end points. It is plausible that this can be done with some smooth additive modifier functions, but explicit examples may not be easy to find.