Approximate utility

Approximating arbitrage or a perfect hedge in a limit implies approximation in utility (formally, expected utilities are continuous in $L^2$), if and only if the utility function is bounded above and below by quadratics. We characterize when some special utility functions are continuous for all random consumptions with a common lower bound. Only linear, quadratic, and, in some cases, translated SAHARA utilities are continuous if there is no lower bound. We also characterize when a utility function defined on a convex subset can be extended to an $L^2$-continuous nondecreasing and concave utility function on all of $R$.