# Questions tagged [semicontinuous-functions]

Tag for questions about upper-semicontinuous and lower-semicontinuous functions.

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### Rudin's RCA Theorem 7.21

Theorem 7.21 states: If $f:[a,b]\to\mathbb{R}$ is differentiable at every point of $[a,b]$ and $f'\in L^1$ on $[a,b]$, then $f(x)-f(a)=\int_a^x f'(t)dt$ for all $x\in[a,b]$. From a very early theorem, ...
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### Are hyperplanes Upper and Lower semicontinous correspondence?

Let $\Phi:\mathbb{R}^2 \to P(\mathbb{R}^2)$ where $P(⋅)$ is a power set, defined by $\Phi(d) = \{x\in\mathbb{R}^2: d^T x =0\}$ where $d^T$ is the transpose of vector $d$. So, it is clear that for a ...
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### A supremum/integral inequality

Let $I$ be an arbitrary index set and for each $i\in I$ let $f_i:\mathbb{R}^n\to [0,\infty)$ be a measurable function. For each $i\in I$ and $n\in \mathbb{N}$ let $f_i^n:\mathbb{R}^n\to [0,\infty)$ be ...
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### Restriction of global sections to the fiber without semicontinuity

Let $f:X\to Y$ be a projective morphism of noetherian schemes and let $\mathcal{F}$ be a coherent sheaf on $X$ which is flat over $Y$. If you read Hartshorne chapter 3 section 12 or Vakil chapter 25, ...
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### Confused about lower semicontinuous functions

LSC is basically : For a function f to be lower semicontinuous at a means that if x is near a then f(x) is greater than or equal to f(a), or at least nearly so. But in the given graph, f is not LSC ...
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### upper semicontinuous definitions

I'm learning about upper (lower) semicontinuous and saw $3$ defintions for this: Definition 1: The function $f: A\subset\mathbb{R}^n\to\mathbb{R}$ is called upper semicontinuous(resp.: lower ...
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### How do we know that the maximum/minimum is attainable in the definition of viscosity solution?

When I was learning the definition of viscosity solution, I had the following confusion： The definition of viscosity solution in <Continuous-time stochastic control and optimization with financial ...
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### In a regular top space, is it true that for every $x \in X$ and closed set $C \subseteq X$, $\exists$ open nbh of x $G$ s.t $\overline{G}\subset C^C$?

As part of a bigger problem the following argument is applied: Let $X$ be a regular topological space I is a lower semicontinuous function $X \rightarrow [0,+\infty]$. (...) Let $x \in X$ be fixed and ...
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### A question about a normal lower semicontinuos function [closed]

Theorem: A lower semicontinuos function on a topological space $X$ is normal if and only if for each real number $\lambda$, $\left\{ x\in X:f\left( x\right) <\lambda \right\}$ is a union of ...
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### Image of lower semi-continuous function on lower-bounded sets

Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a lower semi-continuous function. Prove or disprove that if $X \subset \mathbb{R}$ has a lower bound, then $f(X)$ also has a lower bound. My attempt: So far, ...
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This is a follow-up question to this question. I read a more elegant looking proof in [1] Proposition 8.44, p208. So it goes like this. Proposition If $x_n \rightharpoonup x$ then $$\left\Vert x \... • 664 0 votes 0 answers 17 views ### Show that \mathbb{I}_A is a lower semi continuous function for an open set A in \mathbb{R} Suppose A is an open set in \mathbb{R}. Show that \mathbb{I}_A is a lower semi continuous function where, a function, f:\mathbb{R}\rightarrow\mathbb{R} is said to be lower semi continuous at ... • 2,224 0 votes 2 answers 119 views ### F is lower semicontinuous \iff F(x)=\sup_{r>0}\inf_{y\in B(x,r)}F(y) for all x\in X Let (X,d) be a metric space. F: X\to \overline{\mathbb{R}}. The definitions that I have to use are: (1) F is sequentially lower semicontinuous if for all sequences (x_n)_n \subseteq X s.t  x_n\... • 3,152 0 votes 1 answer 58 views ### (X,d) be a metric space. F: X\to \overline{\mathbb{R}}. F is sequentially lower semicontinuous iff F is lower semicontinuous Let (X,d) be a metric space. F: X\to \overline{\mathbb{R}}. F is sequentially lower semicontinuous iff F is lower semicontinuous The definitions that I have to use are: (1) F is sequentially ... • 3,152 1 vote 1 answer 158 views ### How do I prove these results involving the liminf and the inf ? ( part of a proof on sequential lower semicontinuity of lower semicontinuous envelope) I am trying to prove equation (6.3) in the lemma below. This is part of a course in calculus of variations, but that is irrelevant here, this is actually a question about the liminf and the inf of ... • 3,152 2 votes 0 answers 62 views ### Brezis' Exercise 4.10.1 and 4.10.2 I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Let (\Omega, \mathcal F, \mu) be a measure space with \mu(\Omega) < \infty. Let p \in [1, \infty) and j:\mathbb R \... • 17.6k 1 vote 0 answers 51 views ### Prove that \operatorname{cl}(\operatorname{epi}(f))\subset\operatorname{epi}(\operatorname{cl}f). Let f:\mathbb{R}\to\mathbb{R}.$$(\operatorname{cl}f)(x):=\liminf_{x'\to x}f(x')$$Prove that \operatorname{cl}(\operatorname{epi}(f))\subset\operatorname{epi}(\operatorname{cl}f). \forall x,\... • 925 1 vote 2 answers 109 views ### Is there a lower semicontinuous function f:[0,1] \to [0,1]  such that the set \{t\in (0,1] : \liminf_{s\nearrow t} f(s) > f(t)\} is uncountable? Let f:[0,1] \to [0,1]  be a lower semicontinuous function. I am interested in the set$$ S:= \left\{t\in (0,1] : \liminf_{s\nearrow t} f(s) > f(t)\right\}$$Is there an example for which this is ... • 4,122 2 votes 1 answer 83 views ### Relation between a convex lower semi-continuous function and its integral functional. I'm looking for a reference for the following theorem. In some multidimensional calculus of variations script I found the following theorem. Theorem: Let \Omega \subset \mathbb{R}^n be open and ... • 41 4 votes 1 answer 250 views ### Definition of upper semi-continuous functions: limsup or liminf? Notation: \{f\geq c\} stands for \{x\in x: fx\geq c\}. The standard definition of an upper semi-continuous function f:X\to \bar{ \mathbb R} is: For each c in \mathbb R, \{f\geq c\} is ... 1 vote 1 answer 361 views ### Different definition of lower semi-continuity I am struggling to understand how different definitions of lower semi-continuity compare to each other. I am mostly concerned with that concept regarding closed convex functions. Therefore I am ... • 353 0 votes 1 answer 56 views ### Is this function lower semicontinuous? Prove or disprove that f:\mathbb{R^2}\to \mathbb{R} with$$f(x, y) =\begin{cases} x^2+y^2-1 & \text{ if } (x,y) \in\{(z_1,z_2)\in\mathbb{R}^2:\sqrt{z_1^2+z_2^2}\leq 1\} \\ x^2+y^2 & \text{ ...
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If $f,g:[0,\infty)\mapsto [0,\infty)$ and $f<g$ are upper and lower semicontinuous nondecreasing functions, is there some continuous nondecreasing function $h:[0,\infty)\mapsto [0,\infty)$ such ...