Semi-continuity

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In mathematical analysis, semi-continuity (or semicontinuity) is a property of real-valued functions that is weaker than continuity. A real-valued function f is upper 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 f(x0). If "less than" is replaced by "greater than", the function is called lower semi-continuous at x0.

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[edit] Examples

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

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} x^2 & \mbox{if } 0 \le x < 1,\\ 2 & \mbox{if } x = 1, \\ 1/2 + (1-x) & \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.

[edit] Formal definition

Suppose X is a topological space, x0 is a point in X and f : X → R is a 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

\limsup_{x \to x_{0}} f(x) \leq f(x_{0})

where lim sup is the limit superior (of 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

\liminf_{x \to x_{0}} f(x) \geq f(x_{0})

where lim inf is the limit inferior (of 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.

[edit] Properties

A function is continuous at x0 if and only if it is upper and lower semi-continuous there.

If f and g are two 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 interval [a, b]) and f : C → R is upper semi-continuous, then f has a maximum on C. The analogous statement for lower semi-continuous functions and minima is also true. (See the article on the extreme value theorem for a proof.)

Suppose fn : X → R is a lower semi-continuous function for every natural number n, and

f(x) := sup {fn(x) : n in N} < ∞

for every x in X. Then f is lower semi-continuous. Even if all the fn 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.

[edit] 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. 

[edit] See also