  One of the crucial problems in mathematical finance is to mitigate the risk
of a financial position by setting up hedging positions of eligible financial
securities. This leads to focusing on set-valued maps associating to any
financial position the set of those eligible payoffs that reduce the risk of
the position to a target acceptable level at the lowest possible cost. Among
other properties of such maps, the ability to ensure lower semicontinuity and
continuous selections is key from an operational perspective. It is known that
lower semicontinuity generally fails in an infinite-dimensional setting. In
this note we show that neither lower semicontinuity nor, more surprisingly, the
existence of continuous selections can be a priori guaranteed even in a
finite-dimensional setting. In particular, this failure is possible under
arbitrage-free markets and convex risk measures.
