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 do not rapidly jump upwards. If they don't rapidly jump downwards, the function is called lower semi-continuous at x0.
Examples
Image:Upper semi.png Consider the function 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.
Image:Lower semi.png The floor function <math>f(x)=\lfloor x \rfloor</math>, which returns the greatest integer smaller than a given <math>x</math>, is everywhere upper semi-continuous. Similarily the ceiling function <math>f(x)= \lceil x \rceil</math> is lower semi-continuous.
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
- lim sup x → x0f(x) ≤ f(x0)
(see limit superior and limit inferior for the definition of lim sup). The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain. Then {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
- lim inf x → x0 f(x) ≥ f(x0)
(see limit superior and limit inferior for the definition of lim inf). The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain. Then {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.
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.
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.
The indicator function of any open set is lower semicontinuous. The characteristic function of a closed set is upper semicontinuous.de:Halbstetigkeit