Semicontinuity

Remarks

The function $f$ is upper semicontinuous if and only if $\set{x : f(x) < y}$ is open for every real $y$. It is lower semicontinuous if and only if $\set{x : f(x) > y}$ is open for every real $y$. In terms of limits, $f$ is upper semicontinuous at $x_0$ if and only if $\limsup_{x \to x_0} f(x) \le f(x_0)$, and lower semicontinuous at $x_0$ if and only if $\liminf_{x \to x_0} f(x) \ge f(x_0)$. A real-valued $f$ is continuous at $x_0$ if and only if it is both upper and lower semicontinuous at $x_0$.

Example: the floor function $\floor{\cdot} \colon \R \to \R$ is upper semicontinuous. At an integer $n$ and for $y > n$, the neighbourhood $(n - 1, n + 1)$ works, since $\floor{x} \le n < y$ there. At a non-integer, the floor function is locally constant.

Non-example: the indicator function $f$ of $[0, 1]$ in $\R$ is not lower semicontinuous at $0$. Here $f(0) = 1$; for $y = 1/2$, every neighbourhood of $0$ contains some $x < 0$ with $f(x) = 0 < y$.