Semi-continuity

In mathematics, a real-valued function is called upper continuous (or semi-continuous from above ) at a point when the function values ​​for arguments close to not jump starting at from the top. If the function values ​​do not jump down, then the function is called lower semicontinuous in (or semi-continuous from below).

Definition

Be a topological space, and in a real-valued function. ie in above continuous if there exists for each one around, so that applies in all. If a room in which each follow a continuous function is also continuous, as a metric space, clearly above then steadily, if

Above is continuous on a subset of, if it is continuous at each point above. Is this the whole topological space, so called upper continuous.

The analogy in point lower semicontinuous if for each there exists a neighborhood such that for all. If a room in which each follow a continuous function is also continuous, as a metric space, then if and only lower semicontinuous in, if

Is called lower semicontinuous on a subset of, if it is lower semicontinuous at each point. Is this the whole topological space, so called lower semicontinuous.

Connection of the two semi-continuity concepts: The function is exactly above then steadily in or on when is lower semicontinuous in or on.

Examples

The function for <0 and ≥ 0 is upper continuous, but not lower semicontinuous at = 0 because it goes with the arguments in the negative direction from the leading 0, then the function values ​​suddenly jump from 1 to 0 down, but they do not jump after above, no matter where you go away.

The floor function is upper continuous, because it behaves at each integer as the function just described.

Properties

A function is continuous at if and only if there is semicontinuous from above and from below.

Are and two in above continuous functions, then their sum in upper continuous. If both functions are non-negative in a neighborhood of, then the product is above steadily. Multiplying a positive continuous function above with a negative real number gives a lower semi continuous function.

Is a compact set (for example, a closed interval of real numbers ) and above continuous, then has a maximum. The same is true for a lower semi continuous function and its minimum.

Are the functions of lower semicontinuous (for all out ) and their supremum

Smaller than ∞ for each in, then lower semicontinuous. Even if all of them are continuous, but does not have to be continuous.

Alternative description

Above continuous and semi- continuous functions can be regarded as continuous functions by a suitable choice of a topology, and thus, can be some of the properties directly from general statements derived from the topology.

Is a topology. Be a topological space. A function is just above then steadily when the map is continuous.

For under semi- continuous functions using analog topology.