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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. A extended 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.

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

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

$f\left(x\right) = begin\left\{cases\right\}$
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
$f\left(x\right) = begin\left\{cases\right\}$
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.

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

$limsup_\left\{xto x_0\right\} f\left(x\right)le f\left(x_0\right)$

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

$liminf_\left\{xto x_0\right\} f\left(x\right)ge f\left(x_0\right)$

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 [ab]) 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.,

$f\left(x\right)=sup_\left\{iin I\right\}f_i\left(x\right),qquad xin X.$

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.

References

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