Satisfaction of either of the independence axiom, or its less stringent counterpart, `smoothness of utility functions' is necessary condition for robustness of applications of expected utility theory to modeling of choice under uncertainty. This study arrives at a general equilibrium mathematical condition for inferring of, simultaneously, violations of each of the independence axiom or smoothness of utility functions, conditions. The mathematical condition shows functions that are not `well defined', equivalently, not `one-to-one' induce violations of the independence axiom. In stated respect, artifice of truncation of concave functions at first encounter of a zero derivative induces a function that is not well defined. With focus on stock markets, the mathematical condition directly implies non-robustness of either of concave or strictly convex functions to modeling of choice under uncertainty. For concreteness, in presence of satisfaction of ordering, continuity, and independence axioms, regardless, adoption of concave utility functions for modeling of choice under uncertainty embeds several reinforcing endogeneities, namely pre-knowledge of the optimum, dichotomization of expectations of positive returns from information, and dependence of expectations of positive returns on increase to risk aversion parameters of economic agents. While strictly convex functions are well defined, contrary to economics of stock markets, adoption of strictly convex utility functions implies existence of numeraire assets, that is, parameterization of stock markets by First Order Stochastic Dominance, as such, arrival at a contradiction. Study findings reiterate importance of searches for new approaches to modeling of choice under uncertainty.