Upper semicontinuity ($\{x:f(x)<a\}\text{ is open for every }a\in \mathbb{R}$)

A real or extended-real function is upper semicontinuous when every strict sublevel set is open. Equivalently its negative is lower semicontinuous. An arbitrary infimum of continuous functions is upper semicontinuous, even when the infimum is over an uncountable family; its strict sublevel sets are unions of the corresponding open sublevel sets.