Let $f:\mathbb{R}\rightarrow[-\infty,\infty]$ be a continuous function. The convex conjugate of $f$ is:
$$f^*(p) := \sup_{x\in\mathbb{R}}\{px-f(x)\}~.$$
Furthermore, let us define the subderivative $\partial f(a)$ of $f$ at $a$:
$$\partial f(a) := \{y\in[-\infty,\infty]: f(x)-f(a)\ge y(x-a)\}~.$$
I found out that for $f(x)=|x|$:
$$ f^*(p) = \begin{cases} 0 & \text{for } |p|\le 1\\ \infty &\text{else}.\end{cases} $$
How can we show that?