We prove that if $f\,:\,{{\mathbb{R}}^{N}}\,\to \,\overline{\mathbb{R}}$ is quasiconvex and $U\,\subset \,{{\mathbb{R}}^{N}}$ is open in the density topology, then $\underset{U}{\mathop{\sup }}\,f=\text{ess}\,\underset{U}{\mathop{\sup }}\,f$, while ${{\inf }_{U}}\,f\,=\,\text{ess}\,{{\inf }_{U}}\,f$ if and only if the equality holds when $U\,\subset \,{{\mathbb{R}}^{N}}$. The first (second) property is typical of $\text{lsc}\,\text{(usc)}$ functions, and, even when $U$ is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.

This property ensures that the pointwise extrema of $f$ on any nonempty density open subset can be arbitrarily closely approximated by values of $f$ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.