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).

Contents

Examples

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.

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 f(x)=\lfloor x \rfloor, which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function f(x)=\lceil x \rceil is lower semi-continuous.

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

f(x) = \begin{cases} 1 & \mbox{if } x < 1,\\ 2 & \mbox{if } x = 1,\\ 1/2 & \mbox{if } x > 1, \end{cases}

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

f(x) = \begin{cases} \sin(1/x) & \mbox{if } x \neq 0,\\ 1 & \mbox{if } x = 0, \end{cases}

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

Let (X,\mu) be a measure space and let L^%2B(X,\mu) denote the set of positive measurable functions endowed with the topology of \mu-almost everywhere convergence. Then the integral, seen as an operator from L^%2B(X,\mu) to [-\infty,%2B\infty] is lower semi-continuous. This is just Fatou's lemma.

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. For the particular case of a metric space, this can be expressed as

\limsup_{x\to x_{0}} f(x)\le f(x_0)

where lim sup is the limit superior (of the function f at point x0). (For non-metric spaces, an equivalent definition using nets may be stated.)

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

\liminf_{x\to x_0} f(x)\ge f(x_0)

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; alternatively, a function is lower semi-continuous if and only if all of its lower levelsets {x ∈ X : f(x) ≤ α} are closed. Lower level sets are also called sublevel sets or trenches.[1]

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

f(x)=\sup_{i\in I}f_i(x),\qquad x\in X.

Then f is lower semi-continuous. Even if all the fi 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

  1. ^ Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming (Series A) (Berlin, Heidelberg: Springer) 90 (1): pp. 1-25. doi:10.1007/PL00011414. ISSN 0025-5610. MR1819784. 

See also