Consider the following function
$$f(x)= \begin{cases} e^{x/2} \qquad x \leq 0 \\ x + e^{-x} \qquad x >0 \end{cases}$$
I need to show it's Lipschitz with constant $1$: $|T(x)-T(y)|<|x-y|$. For $x,y >0$ and $x,y \leq 0$ I can prove it easily.
But, if $x>0$ and $y<0$, then $$|T(x) - T(y)| = |e^{x/2} - y - e^{-y}|$$ and I don't know how to move from here. Any hint?
If $x>0$ and $y<0$, then by the cases for $x,y\ge0$ and $x,y\le0$ we have $$|T(x)-T(y)|\le|T(x)-T(0)|+|T(0)-T(y)|\le |x-0|+|0-y|=x-y.$$