Let $X$ be a scalar random variable having $F_X$ as its cdf. Let $p_t$ be the function defined by $p_t(u)=u(t-\mathbf 1_{u\lt0})$ where $\mathbf 1$ denotes the indicator function. Let $$q_t=\arg\min E\left[p_t(X-a)\right].$$
Show that $$F_X(q_t)=t.$$