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Semicontinuous-functions
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Convex closed and unclosed functions and (lower semi)continuity
September 13th, 2020
real-analysis
semicontinuous-functions
convex-analysis
continuity
Show that lower semi-continuous function attains it's minimum. (Proof verification) (By contradiction)
August 1st, 2022
real-analysis
semicontinuous-functions
upper-lower-bounds
proof-verification
Sum of two upper semi continuous functions is upper semi continuous
August 1st, 2022
semicontinuous-functions
real-analysis
Prove by definition that every upper semi-continuous function can be expressed as infimum of a sequence of continuous functions.
December 8th, 2020
metric-spaces
semicontinuous-functions
supremum-and-infimum
general-topology
Composition of lower semicontinuous function with continuous function is lower semicontinuous
August 1st, 2022
semicontinuous-functions
continuity
real-analysis
function-and-relation-composition
Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions
August 1st, 2022
analysis
general-topology
semicontinuous-functions
Pointwise infimum of an arbitrary collection of upper semicontinuous functions is upper semicontinuous
October 10th, 2020
supremum-and-infimum
semicontinuous-functions
general-topology
real-analysis
Maximum of a upper semicontinuous function
August 1st, 2022
real-analysis
analysis
semicontinuous-functions
Show that if $f^{-1}((\alpha, \infty))$ is open for any $\alpha \in \mathbb{R}$, then $f$ is lower-semicontinuous.
May 9th, 2020
general-topology
semicontinuous-functions
real-analysis
Show a function is continuous if and only if it is both upper and lower semi-continuous
August 1st, 2022
semicontinuous-functions
continuity
measure-theory
real-analysis
analysis
Lower Semicontinuity Concepts
August 1st, 2022
functional-analysis
general-topology
semicontinuous-functions
examples-counterexamples
banach-spaces
Show that every upper semi-continuous real function is measurable
August 1st, 2022
measure-theory
semicontinuous-functions
real-analysis
Lower semicontinuous function as the limit of an increasing sequence of continuous functions
August 1st, 2022
semicontinuous-functions
real-analysis
To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous
August 1st, 2022
general-topology
supremum-and-infimum
semicontinuous-functions
continuity
Upper semi continuous, lower semi continuous
August 1st, 2022
real-analysis
semicontinuous-functions
Any lower semicontinuous function $f: X \to \mathbb{R}$ on a compact set $K \subseteq X$ attains a min on $K$.
August 8th, 2020
semicontinuous-functions
measure-theory
Approximation of Semicontinuous Functions
August 1st, 2022
approximation
semicontinuous-functions
real-analysis
approximation-theory
What is the intuition for semi-continuous functions?
August 1st, 2022
semicontinuous-functions
real-analysis
Show the iff statement of lower semicontinuous
August 1st, 2022
semicontinuous-functions
real-analysis
Equivalence of definitions for upper semicontinuity
August 1st, 2022
semicontinuous-functions
real-analysis
Show that the supremum of a collection of lower semicontinuous function is lower semicontinuous
August 1st, 2022
real-analysis
continuity
semicontinuous-functions
Subset of the preimage of a semicontinuous real function is Borel
August 1st, 2022
real-analysis
semicontinuous-functions
measure-theory
borel-sets