Encyclopedia > Semi-continuous

  Article Content

Semi-continuous

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

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

Imagine that you are scanning a certain scenery with your eyes and record the distance to the viewed object at all times. This yields a lower semi-continuous function which in general is not upper semi-continuous (for instance if you focus on the edge of a table).

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 supxx0 f(x) ≤ f(x0).
The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain.

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 infxx0 f(x) ≥ f(x0).
The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain.

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-, 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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
List of rare diseases starting with A

... Angiosarcoma of the scalp[?] Angiostrongyliasis[?] Angiotensin renin aldosterone hypertension[?] Anguillulosis[?] Aniridia absent patella[?] Anirid ...

 
 
 
This page was created in 43.5 ms