Semi-continuity

Remarks

For $X \subseteq \R$, upper semicontinuity at $x_0$ means $\limsup_{x \to x_0} f(x) \le f(x_0)$, and lower semicontinuity means $\liminf_{x \to x_0} f(x) \ge f(x_0)$. The function $f$ is upper semicontinuous if and only if the set $\set{x \in X : f(x) < y}$ is open for each real $y$. It is lower semicontinuous if and only if $\set{x \in X : f(x) > y}$ is open for each real $y$. The function $f$ is upper semicontinuous if and only if $-f$ is lower semicontinuous. A real function is continuous at $x_0$ if and only if it is both upper and lower semicontinuous at $x_0$.

Example: the indicator function $f = \mathbf 1_{[0, 1]}$ on $\R$ is upper semicontinuous. The set $\set{f < y}$ is empty for $y \le 0$, is $\R \setminus [0, 1]$ for $0 < y \le 1$, and is $\R$ for $y > 1$, and each of these is open.

Non-example: the same function is not lower semicontinuous at $0$. Take $y = 1/2 < 1 = f(0)$; each neighbourhood of $0$ contains a point $x < 0$ with $f(x) = 0 < y$.