PlanetMath: semicontinuous

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semicontinuous

(Definition)

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

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

In other words, is lower semicontinuous, if is continuous with respect to the topology for $\mathbb{R}^*$ containing $\emptyset$ and open sets

$\displaystyle U(\alpha) = (\alpha,\infty], \quad \quad \alpha\in \mathbb{R}\cup \{-\infty\}.$

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

A function $f\colon X\to \mathbb{R}^*$ is continuous if and only if it is lower and upper semicontinuous.
Let be the characteristic function of a set $\Omega\subseteq X$ . Then 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.
On $\mathbb{R}$ , the function for $x\neq 0$ and , is not semicontinuous. This example illustrate how semicontinuous "at infinity".

Properties

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

Restricting to a subspace preserves semicontinuity.
Suppose is upper (lower) semicontinuous, is a topological space, and $\Psi\colon A\to X$ is a homeomorphism. Then $f\circ\Psi$ is upper (lower) semicontinuous.
Suppose 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.
is lower semicontinuous if and only if 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.

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"semicontinuous" is owned by bwebste. [ full author list (5) | owner history (2) ]

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Also defines:

lower semicontinuous, upper semicontinuous, lower semi-continuous, upper semi-continuous

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Cross-references: homeomorphism, preserves, subspace, closed, characteristic function, continuous at, infinity, neighborhoods, numbers, usual topology, coarser, union, continuous, open set, extended real numbers, function, topological space
There are 10 references to this entry.

This is version 10 of semicontinuous, born on 2003-10-15, modified 2007-05-09.
Object id is 4844, canonical name is LowerSemicontinuous.
Accessed 20100 times total.

Classification:

AMS MSC:

26A15 (Real functions :: Functions of one variable :: Continuity and related questions )

Pending Errata and Addenda

None.[ View all 1 ]

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