semi-continuity
Sign in to savethumb|right|An upper semicontinuous function that is not lower semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right). thumb|right|A lower semicontinuous function that is not upper semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right).
Article
23 sectionsContents
- Definitions
- Upper semicontinuity
- Lower semicontinuity
- Examples
- Properties
- Binary operations on semicontinuous functions
- Optimization of semicontinuous functions
- Other properties
- Semicontinuity of set-valued functions
- Upper and lower semicontinuity
- Inner and outer semicontinuity
- Hulls
- Applications
- Calculus of variations
- Existence of saddle points
- Dimension
- Algebraic geometry
- Descriptive set theory
- Dynamical systems
- See also
- Notes
- References
- Bibliography
thumb|right|An upper semicontinuous function that is not lower semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right). thumb|right|A lower semicontinuous function that is not upper semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right).
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). Briefly, a function on a domain X is lower semi-continuous if its epigraph \{(x,t)\in X\times\R : t\ge f(x)\} is closed in X\times\R, and upper semi-continuous if -f is lower semi-continuous.