# semicontinuous

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. 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. 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).

## 0.0.1 Examples

1. 1.

A function $f\colon X\to\mathbb{R}^{*}$ is continuous if and only if it is lower and upper semicontinuous.

2. 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 $\mathbb{Q}$ is not semicontinuous.

3. 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”.

## 0.0.2 Properties

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

1. 1.

Restricting $f$ to a subspace preserves semicontinuity.

2. 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. 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. 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.
Title semicontinuous Semicontinuous1 2013-03-22 14:00:16 2013-03-22 14:00:16 bwebste (988) bwebste (988) 13 bwebste (988) Definition msc 26A15 lower semicontinuous upper semicontinuous lower semi-continuous upper semi-continuous