Let be a metric space (or more generally a topological space). A function is lower semicontinuous if
is open in for all .
- If is lower semicontinuous and the is lower semicontinuous.
- If is a topological space and is any open set, then is lower semicontinuous.
- If are lower semicontinuous, then is lower semicontinuous.
- If is a locally compact Hausdorff space, and is lower semicontinuous, then where denotes the space of all continuous functions on with compact support.
- If is an convergent sequence in converging to some , then .
- If is continuous, then it is lower semicontinuous. 
- In the case that , is Borel-measurable. 
- If is a collection of lower semicontinuous functions from to , then the function is lower semicontinuous.
Lower Semicontinuous Envelope
Given any bounded function , the lower semicontinuous envelope of , denoted is the lower semicontinuous function defined as
- ↑ 1.0 1.1 1.2 Craig, Katy. MATH 201A HW 1. UC Santa Barbara, Fall 2020.
- ↑ 2.0 2.1 2.2 2.3 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §7.2
- ↑ Craig, Katy. MATH 201A HW 4. UC Santa Barbara, Fall 2020.
- ↑ Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.