General topology Continuous maps Concept
Semicontinuity
Summary
Semicontinuity is a one-sided form of continuity for functions with values in the extended real line. A function is upper semicontinuous when its value at a point may lie above the nearby values but not below them. It is lower semicontinuous in the opposite case. A function is continuous exactly when it is both. Lower semicontinuous functions attain their minimum on nonempty compact spaces, which makes them basic in optimisation and the calculus of variations.
Where this leads
16 pages build on this one.
- Phragmén–Lindelöf principle for subharmonic functions
- Superharmonic function
- Weak lower semicontinuity
- Proximal operator
- Maximum principle for subharmonic functions
- Sub-mean value property
- Plurisubharmonic function
- Ekeland's variational principle
- Grauert's semicontinuity theorem
- Ky Fan minimax inequality
- Relaxation of a variational problem
- Hadamard three-circle theorem for subharmonic functions
- Lindelöf's maximum principle
- Rate function
- Fenchel–Moreau theorem
- Hartogs lemma for subharmonic functions
Further reading
About this page
| Tag | RN4G8SGP6P |
|---|---|
| URI | m2://RN4G8SGP6P |
| Kind | concept |
| Writing | W0 stub |
| MSC | 54C08, 26A15 |
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Used by
- Superharmonic function
- Weak lower semicontinuity
- Proximal operator
- Maximum principle for subharmonic functions
- Sub-mean value property
- Plurisubharmonic function
- Ekeland's variational principle
- Grauert's semicontinuity theorem
- Ky Fan minimax inequality
- Relaxation of a variational problem
- Rate function
- Fenchel–Moreau theorem