Suppose X is a topological spaceMathworldPlanetmath, and f is a function from X into the extended real numbers *; f:X*. Then:

  1. 1.

    If f-1((α,])={xXf(x)>α} is an open set in X for all α, then f is said to be lower semicontinuous.

  2. 2.

    If f-1([-,α))={xXf(x)<α} is an open set in X for all α, then f is said to be upper semicontinuous.

In other words, f is lower semicontinuous, if f is continuousMathworldPlanetmathPlanetmath with respect to the topologyMathworldPlanetmath for * containing and open sets


It is not difficult to see that this is a topology. For example, for a union of sets U(αi) we have iU(αi)=U(infαi). Obviously, this topology is much coarserPlanetmathPlanetmath than the usual topology for the extended numbers. However, the sets U(α) can be seen as neighborhoods of infinityMathworldPlanetmath, so in some sense, semicontinuous functions are ”continuous at infinity” (see example 3 below).

0.0.1 Examples

  1. 1.

    A function f:X* is continuous if and only if it is lower and upper semicontinuous.

  2. 2.

    Let f be the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of a set ΩX. Then f is lower (upper) semicontinuousPlanetmathPlanetmath if and only if Ω is open (closed). This also holds for the function that equals in the set and 0 outside.

    It follows that the characteristic function of is not semicontinuous.

  3. 3.

    On , the function f(x)=1/x for x0 and f(0)=0, is not semicontinuous. This example illustrate how semicontinuous ”at infinity”.

0.0.2 Properties

Let f:X* be a function.

  1. 1.

    Restricting f to a subspaceMathworldPlanetmath preserves semicontinuity.

  2. 2.

    Suppose f is upper (lower) semicontinuous, A is a topological space, and Ψ:AX is a homeomorphism. Then fΨ is upper (lower) semicontinuous.

  3. 3.

    Suppose f is upper (lower) semicontinuous, and S:** is a sense preserving homeomorphism. Then Sf is upper (lower) semicontinuous.

  4. 4.

    f is lower semicontinuous if and only if -f is upper semicontinuous.


  • 1 W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.
  • 2 D.L. Cohn, Measure Theory, Birkhäuser, 1980.
Title semicontinuous
Canonical name Semicontinuous1
Date of creation 2013-03-22 14:00:16
Last modified on 2013-03-22 14:00:16
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 13
Author bwebste (988)
Entry type Definition
Classification msc 26A15
Defines lower semicontinuous
Defines upper semicontinuous
Defines lower semi-continuous
Defines upper semi-continuous