Suppose $X$ is a topological space, and $f$ is a function from $X$ into the extended real numbers $\mathbb{R}^*$ ; $f:X\to \mathbb{R}^*$ . Then:

1. If $f^{-1}((\alpha,\infty])=\{x\in X \mid f(x) >\alpha\}$ is an open set in $X$ for all $\alpha\in \mathbb{R}$ , then $f$ is said to be lower semicontinuous.
2. If $f^{-1}([-\infty,\alpha))=\{x\in X \mid f(x) <\alpha\}$ is an open set in $X$ for all $\alpha\in \mathbb{R}$ , then $f$ is said to be upper semicontinuous.

In other words, $f$ is lower semicontinuous, if $f$ is continuous with respect to the topology for $\mathbb{R}^*$ containing $\emptyset$ and open sets $$ U(\alpha) = (\alpha,\infty], \quad \quad \alpha\in \mathbb{R}\cup \{-\infty\}. $$ It is not difficult to see that this is a topology. For example, for a union of sets $U(\alpha_i)$ we have $\cup_i U(\alpha_i)=U(\inf \alpha_i)$ . Obviously, this topology is much coarser than the usual topology for the extended numbers. However, the sets $U(\alpha)$ can be seen as neighborhoods of infinity, so in some sense, semicontinuous functions are "continuous at infinity" (see example 3 below).

Examples

1. A function $f\colon X\to \mathbb{R}^*$ is continuous if and only if it is lower and upper semicontinuous.
2. Let $f$ be the characteristic function of a set $\Omega\subseteq X$ . Then $f$ is lower (upper) semicontinuous if and only if $\Omega$ is open (closed). This also holds for the function that equals $\infty$ in the set and $0\,$ outside.

   It follows that the characteristic function of $\Q$ is not semicontinuous.

3. On $\mathbb{R}$ , the function $f(x)=1/x$ for $x\neq 0$ and $f(0)=0$ , is not semicontinuous. This example illustrate how semicontinuous "at infinity".

Properties

Let $f\colon X\to \mathbb{R}^*$ be a function.

1. Restricting $f$ to a subspace preserves semicontinuity.
2. Suppose $f$ is upper (lower) semicontinuous, $A$ is a topological space, and $\Psi\colon A\to X$ is a homeomorphism. Then $f\circ\Psi$ is upper (lower) semicontinuous.
3. Suppose $f$ is upper (lower) semicontinuous, and $S\colon \mathbb{R}^*\to \mathbb{R}^*$ is a sense preserving homeomorphism. Then $S\circ f$ is upper (lower) semicontinuous.
4. $f$ is lower semicontinuous if and only if $-f$ is upper semicontinuous.

References

1

W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.

2

D.L. Cohn, Measure Theory, Birkhäuser, 1980.