 Solution

On each open half-line the sample objective is a utility of a linear gain: $U(\theta (X-\epsilon ))$ on the positive half-line, and $U(\theta (X+\epsilon ))$ on the negative half-line. On a compact subinterval wholly inside either half-line, the secant domination argument from the frictionless part ii applies verbatim, using finiteness of $G$ at four surrounding points. The dominated convergence theorem therefore gives the derivativesThey are continuous on their respective half-lines. Near zero, the same outer secants of the globally concave sample objective bound its one-sided slopes. Dominated convergence consequently givesHere $\mathbb{E}|X|<\infty$: the frictionless derivative at zero already proves it, since $U^{\prime }(0)>0$. Thus zero generally has a transaction-cost kink; when $\epsilon =0$ the two formulas agree.