= Solution
On each open half-line the sample objective is a utility of a linear gain: $U(\theta(X-\varepsilon))$ on the positive half-line, and $U(\theta(X+\varepsilon))$ 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 \b[the derivatives]
$$
\boxed{G'(\theta)=\begin{cases}\mathbb E[(X-\varepsilon)U'(\theta(X-\varepsilon))],&\theta>0,\\\mathbb E[(X+\varepsilon)U'(\theta(X+\varepsilon))],&\theta<0.\end{cases}}
$$
They 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 gives
$$
G'_+(0)=U'(0)(\mathbb EX-\varepsilon),\qquad G'_-(0)=U'(0)(\mathbb EX+\varepsilon).
$$
Here $\mathbb E|X|<\infty$: the frictionless derivative at zero already proves it, since $U'(0)>0$. Thus zero generally has a transaction-cost kink; when $\varepsilon=0$ the two formulas agree.
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