Semi-continuity

For the notion of upper or lower semicontinuous multivalued function see: Hemicontinuity

In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (lower) semi-continuous at a point x₀ if, roughly speaking, the function values for arguments near x₀ are either close to f(x₀) or less than (greater than) f(x₀).

Examples

An upper semi-continuous function. The solid blue dot indicates f(x₀).

Consider the function f, piecewise defined by f(x) = –1 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semi-continuous at x₀ = 0, but not lower semi-continuous.

A lower semi-continuous function. The solid blue dot indicates f(x₀).

The indicator function of an open set is lower semi-continuous, whereas the indicator function of a closed set is upper semi-continuous. The floor function , which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function is lower semi-continuous.

A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function

is upper semi-continuous at x = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function

is upper semi-continuous at x = 0 while the function limits from the left or right at zero do not even exist.

Formal definition

Suppose X is a topological space, x₀ is a point in X and f : X → R ∪ {–∞,+∞} is an extended real-valued function. We say that f is upper semi-continuous at x₀ if for every ε > 0 there exists a neighborhood U of x₀ such that f(x) ≤ f(x₀) + ε for all x in U. Equivalently, this can be expressed as

where lim sup is the limit superior (of the function f at point x₀).

The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain. A function is upper semi-continuous if and only if {x ∈ X : f(x) < α} is an open set for every α ∈ R.

We say that f is lower semi-continuous at x₀ if for every ε > 0 there exists a neighborhood U of x₀ such that f(x) ≥ f(x₀) – ε for all x in U. Equivalently, this can be expressed as

where lim inf is the limit inferior (of the function f at point x₀).

The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain. A function is lower semi-continuous if and only if {x ∈ X : f(x) > α} is an open set for every α ∈ R.

Properties

A function is continuous at x₀ if and only if it is upper and lower semi-continuous there. Therefore, semi-continuity can be used to prove continuity.

If f and g are two real-valued functions which are both upper semi-continuous at x₀, then so is f + g. If both functions are non-negative, then the product function fg will also be upper semi-continuous at x₀. Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function.

If C is a compact space (for instance a closed, bounded interval [a, b]) and f : C → [–∞,∞) is upper semi-continuous, then f has a maximum on C. The analogous statement for (–∞,∞]-valued lower semi-continuous functions and minima is also true. (See the article on the extreme value theorem for a proof.)

Suppose f_i : X → [–∞,∞] is a lower semi-continuous function for every index i in a nonempty set I, and define f as pointwise supremum, i.e.,

Then f is lower semi-continuous. Even if all the f_i are continuous, f need not be continuous: indeed every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions.

The indicator function of any open set is lower semicontinuous. The indicator function of a closed set is upper semicontinuous.

A function f : Rⁿ→R is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed.

A function f : X→R, for some topological space X, is lower semicontinuous if and only if it is continuous with respect to the Scott topology on R.

References

• Bourbaki, Nicolas (1998). Elements of Mathematics: General Topology, 1–4. Springer. ISBN 0201006367. 
• Bourbaki, Nicolas (1998). Elements of Mathematics: General Topology, 5–10. Springer. ISBN 3540645632. 
• Gelbaum, Bernard R.; Olmsted, John M.H. (2003). Counterexamples in analysis. Dover Publications. ISBN 0486428753. 
• Hyers, Donald H.; Isac, George; Rassias, Themistocles M. (1997). Topics in nonlinear analysis & applications. World Scientific. ISBN 9810225342. 

See also

• Directional continuity
• Semicontinuous multivalued function