Intimate relationships are investigated between connectedness properties of the lower level sets of a real functionf on a topological spaceX and the uniqueness of suitably defined minimizing sets forf. Two distinct theories are presented, the simpler one pertaining to the LE-level sets $$LE_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} $$ and the other to the LT-level sets $$LT_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} .$$ In each theory, a specific notion of minimizing set is defined in such a way that a functionf having connected level sets can have at most one minimizing set. That this uniqueness is not trivial, however, is shown by the converse result that, ifX is Hausdorff and the sets LEα(f) are all compact, then, in each theory,f has a unique minimizing set only if it has connected level sets. The paper concludes by showing that functions with connected LT-level sets arise naturally in parametric linear programming.