If is an open set in for all , then is said to be upper semicontinuous.
It is not difficult to see that this is a topology. For example, for a union of sets we have . Obviously, this topology is much coarser than the usual topology for the extended numbers. However, the sets can be seen as neighborhoods of infinity, so in some sense, semicontinuous functions are ”continuous at infinity” (see example 3 below).
A function is continuous if and only if it is lower and upper semicontinuous.
It follows that the characteristic function of is not semicontinuous.
On , the function for and , is not semicontinuous. This example illustrate how semicontinuous ”at infinity”.
Let be a function.
Suppose is upper (lower) semicontinuous, is a topological space, and is a homeomorphism. Then is upper (lower) semicontinuous.
Suppose is upper (lower) semicontinuous, and is a sense preserving homeomorphism. Then is upper (lower) semicontinuous.
is lower semicontinuous if and only if is upper semicontinuous.
- 1 W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.
- 2 D.L. Cohn, Measure Theory, Birkhäuser, 1980.
|Date of creation||2013-03-22 14:00:16|
|Last modified on||2013-03-22 14:00:16|
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