= Solution
The <indicator functional of a constraint set> restricts the conjugate supremum to $[-1,1]$. Complete the square:
$$
px-\frac{x^2}{2}=\frac{p^2}{2}-\frac{(x-p)^2}{2}.
$$
Its maximizer is $x=p$ when $|p|\leq1$, and the nearer endpoint $x=\operatorname{sign}(p)$ otherwise. The <convex conjugate of a constrained quadratic> is therefore
$$
\boxed{E^*(p)=\begin{cases}\frac12p^2,&|p|\leq1,\\|p|-\frac12,&|p|>1.\end{cases}}
$$
This is the unit-threshold <Huber loss>; both branches and their first derivatives agree at the transition points. The original PDF constrains $|x|\leq1$, including both signs; the damaged TeX loses that absolute value.
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