upper semi-continuous - meaning and definition. What is upper semi-continuous
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  3. upper semi-continuous

What (who) is upper semi-continuous - definition

PROPERTY OF FUNCTIONS WEAKER THAN CONTINUITY
Semicontinuous; Semicontinuity; Semi-continuous; Lower semi-continuous; Upper semi-continuous; Lower semicontinuous; Upper semicontinuous; Semi-continuous function; Semi-continuous mapping; Semicontinuous function; Upper-semicontinuous; Upper semicontinuity; Lower semicontinuity; Upper semi-continuity; Lower semi-continuity
  • A lower semicontinuous function that is not upper semicontinuous. The solid blue dot indicates <math>f\left(x_0\right).</math>
  • An upper semicontinuous function that is not lower semicontinuous. The solid blue dot indicates <math>f\left(x_0\right).</math>

Semi-continuity         
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than f\left(x_0\right).
https://en.wikipedia.org/wiki/Semi-continuity
Continuous function         
  • The graph of a [[cubic function]] has no jumps or holes. The function is continuous.
  • 1=exp(0) = 1}}
  • section 2.1.3]]).
  • 1=''ε'' = 0.5}}.
  • Riemann sphere]] is often used as a model to study functions like the example.
  • The graph of a continuous [[rational function]]. The function is not defined for <math>x = -2.</math> The vertical and horizontal lines are [[asymptote]]s.
  • For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
  • oscillation]].
  • The sinc and the cos functions
  • Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.
  • thumb
FUNCTION SUCH THAT THE PREIMAGE OF AN OPEN SET IS OPEN
Continuity property; Continuous map; Continuous function (topology); Continuous (topology); Continuous mapping; Continuous functions; Continuous maps; Discontinuity set; Noncontinuous function; Discontinuous function; Continuity (topology); Continuous map (topology); Sequential continuity; Stepping Stone Theorem; Continuous binary relation; Continuous relation; Topological continuity; Right-continuous; Right continuous; Left continuous; Left-continuous; C^1; Continuous fctn; Cts fctn; E-d definition; Continuous variation; Continuity space; Continuous space; Real-valued continuous functions; Left-continuous function; Right-continuous function; Left- or right-continuous function; Continuity at a point; Continuous at a point; Continuous extension
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities.
https://en.wikipedia.org/wiki/Continuous_function
continuous function         
  • The graph of a [[cubic function]] has no jumps or holes. The function is continuous.
  • 1=exp(0) = 1}}
  • section 2.1.3]]).
  • 1=''ε'' = 0.5}}.
  • Riemann sphere]] is often used as a model to study functions like the example.
  • The graph of a continuous [[rational function]]. The function is not defined for <math>x = -2.</math> The vertical and horizontal lines are [[asymptote]]s.
  • For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
  • oscillation]].
  • The sinc and the cos functions
  • Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.
  • thumb
FUNCTION SUCH THAT THE PREIMAGE OF AN OPEN SET IS OPEN
Continuity property; Continuous map; Continuous function (topology); Continuous (topology); Continuous mapping; Continuous functions; Continuous maps; Discontinuity set; Noncontinuous function; Discontinuous function; Continuity (topology); Continuous map (topology); Sequential continuity; Stepping Stone Theorem; Continuous binary relation; Continuous relation; Topological continuity; Right-continuous; Right continuous; Left continuous; Left-continuous; C^1; Continuous fctn; Cts fctn; E-d definition; Continuous variation; Continuity space; Continuous space; Real-valued continuous functions; Left-continuous function; Right-continuous function; Left- or right-continuous function; Continuity at a point; Continuous at a point; Continuous extension
A function f : D -> E, where D and E are cpos, is continuous if it is monotonic and f (lub Z) = lub f z | z in Z for all directed sets Z in D. In other words, the image of the lub is the lub of any directed image. All additive functions (functions which preserve all lubs) are continuous. A continuous function has a {least fixed point} if its domain has a least element, bottom (i.e. it is a cpo or a "pointed cpo" depending on your definition of a cpo). The least fixed point is fix f = lub f^n bottom | n = 0..infinity (1994-11-30)

Wikipedia

Semi-continuity

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaystyle x_{0}} if, roughly speaking, the function values for arguments near x 0 {\displaystyle x_{0}} are not much higher (respectively, lower) than f ( x 0 ) . {\displaystyle f\left(x_{0}\right).}

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x 0 {\displaystyle x_{0}} to f ( x 0 ) + c {\displaystyle f\left(x_{0}\right)+c} for some c > 0 {\displaystyle c>0} , then the result is upper semicontinuous; if we decrease its value to f ( x 0 ) − c {\displaystyle f\left(x_{0}\right)-c} then the result is lower semicontinuous.

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.

https://en.wikipedia.org/wiki/Semi-continuity
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