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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 ...
Uday Kumar's user avatar
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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 ...
Pong's user avatar
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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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Property of lower semicontinuous functions

I am trying to prove that f is sequentially lower semicontinuous at x if and only if $f(𝑥)=\sup_{r>0}\inf_{y \in B(x,r)}f(y)$. Following the proof of ($F$ is lower semicontinuous $\iff F(x)=\sup_{...
Sharon Puthuparambil's user avatar
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convex lower semi-continuous function defined on Banach space

I'm working on the Exercise 1 from Chapter 2, in the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haïm Brezis. The exercise states as following: Let $E$ ...
ZENG's user avatar
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Sum of two subharmonic functions is subharmonic.

Definition $:$ Let $G \subseteq \mathbb C$ be a domain. A function $s : G \longrightarrow \mathbb R \cup \{-\infty\}$ is said to be subharmonic if $(1)$ $s$ is upper semicontinuous. $(2)$ given $D \...
Anacardium's user avatar
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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 ...
user17420392's user avatar
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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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Weakly lower-semicontinuous functional on $L^2$

I am interested in the following problem: Is there any function (let's say continuous) $W:\mathbb{R}\to \mathbb{R}$ satisfying $W(x+1)=W(x) \forall x\in \mathbb{R}$ and such that the functional $L^2([...
Pong's user avatar
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Why is the $\Sigma$-product of unit intervals pseudocompact?

Let $A$ be any index set and $X_a$, $a \in A$, be topological spaces. Select any point $x^* = (x_a^*)_{a \in A} \in \prod_{a \in A}X_a$ and denote by $\Sigma_{a \in A}X_a $ the subspace of the product ...
Dmitry's user avatar
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Uniform Convergence of Lagrange function defined for $q-$absolutely continuous curves

In my course on Nonlinear PDEs, we investigate the integral functional $\int_{t_0}^{t_1} L(\dot{\gamma}(t), \gamma(t))dt$, defined on $q$-absolutely continuous curves $\gamma: I \rightarrow \mathbb{R} ...
Len's user avatar
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Lower Semicontinuity of $L^p$ norms with varying exponents

In a previous post (see continuity of $L^p$ norms with respect to $p$) it is shown that in a measure space $(\Omega,\Sigma,\mu)$, if $1\leq p_0\leq p\leq p_1\leq+\infty$, then the function $\Phi\...
Tytiro's user avatar
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If $f(x):=\mu(\overline{B}(x,r))$ show that $f$ is upper semicontinuous

Let $\mu$ be a Radon measure in $\mathbb R^n$, for fixed $r>0$ define $f(x):=\mu(\overline{B}(x,r))$ show that $f$ is upper semicontinuous. Like $\overline{B}(x,r)$ is a closed set then $1_{\...
C L 's user avatar
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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 ...
Jason Li's user avatar
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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 ...
some_math_guy's user avatar
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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 ...
Mehmet Onat's user avatar
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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, ...
Henrique Assumpção's user avatar
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1 answer
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Lim inf with norm and weak convergence[follow up]

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 \...
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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 $...
reyna's user avatar
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$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\...
some_math_guy's user avatar
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(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 ...
some_math_guy's user avatar
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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 ...
some_math_guy's user avatar
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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 \...
Akira's user avatar
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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,\...
qmww987's user avatar
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2 answers
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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 ...
Falrach's user avatar
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2 votes
1 answer
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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 ...
Till S.'s user avatar
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4 votes
1 answer
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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 ...
William Leynoid's user avatar
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 ...
guest1's user avatar
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1 answer
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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{ ...
Uhmm's user avatar
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1 answer
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If f<g are upper and lower semicontinuous nondecreasing functions, is there some continuous nondecreasing function such that f<h<g?

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 ...
Marija's user avatar
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0 answers
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Upper semicontinuity of a function.

Suppose $E\subseteq \mathbb{R}$. A function $f: E\to\mathbb{R}$ is said to be upper semicontinuous at $x_0 \in E$ if for any $\varepsilon>0$, there exists $\delta>0$ such that for any $x\in E$, $...
user136524's user avatar
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1 answer
285 views

Why lower semicontinuity and coercivity implies boundedness of a sequence?

I am reading a proof for Weierstrass' Theorem as follows: For the proof in (2), it says "since $f$ is coercive, ${z_k}$ must be bounded." I am not quite sure how to understand this. Does it ...
fixingmath's user avatar
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1 answer
71 views

Counter example for sum of upper semicontinuous is upper semicontinuous

I needed to find a counter example to "The infinite sum of upper semicontinuous functions is upper semicountinuous". I did find one online, but I wanted to know whether or not mine works. ...
Matt's user avatar
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2 answers
312 views

A sequential characterization of upper semi-continuous function and openness of level set

Question: $(x_n \rightarrow x \implies \limsup_{n \rightarrow \infty } f(x_n) \leq f(x) ) \implies \{x \in X |f(x) < c\} \,\text{is open for any}\, c \in R ?$ I know related questions have been ...
numpynp's user avatar
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2 votes
1 answer
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Approximating a Compact set by approximating its distance function

Let $\emptyset\neq K\subset Y$ be a closed subset of a compact metric space $(X,d)$ such that $K$ has at-least two points and such that $$ d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0. $$ ...
ABIM's user avatar
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From convergence pointwise to convergence of the supremum for semicontinuous functions

Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
G. Panel's user avatar
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To show that a functional is not lower semicontinuous

Let $\Omega\subset \mathbb{R}^2$ be an open and bounded set. Set $M=\{f\in L^2(\Omega):\nabla f\in L^1(\Omega)^2\}$. Define the functional (generalized ROF model) as: $$ J(f)=\begin{cases} \...
creative's user avatar
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Lower Semi Continuous w.r.t weak Topology

I have a question regarding (weakly) lower continuity of some Functionals. Let $\mathbb{H}$ be a reflexive Hilbert space and $A\subseteq \mathbb{H}$ be a closed set in the weak topology of $\mathbb{H}$...
Mathmaxis's user avatar
2 votes
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155 views

Continuity of Kullback-Leibler Divergence/Relative Entropy

Let $Y$ be a measurable subset of an Euclidean space $\mathbb{R}^n$. It is known that the Kullback–Leibler divergence $D_{KL}$ between probability measures on $Y$ is lower-semicontinuous but in ...
FraGarb's user avatar
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2 votes
1 answer
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Does my proof generalize this exercise about continuity of convex lower semi-continuous function?

I'm trying to solve below question (Proposition 0.7.) in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is ...
Akira's user avatar
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Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
Canine360's user avatar
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1 vote
1 answer
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Characterization lower semi continuity

I want to show the characterization of semi-continuity in a general setting, as stated below. Let $X$ a topological space, for $x \in X$ let me denote $\mathcal V(x)$ its neighborhoods. Let $f : X \...
blamethelag's user avatar
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3 votes
1 answer
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the sum of a sequence of lower semicontinuous functions may not be lower semicontinuous?

Let $f_n$ be lower semicontinuous, and $f_n\geq 0$, then $\sum f_n$ is lower semicontinuous. Indeed, for any $c$, $\{x; \sum f_n(x)>c\}=\bigcup_{n=1}^\infty \{x; \sum_{k=1}^n f_k(x)>c\}$. ...
xldd's user avatar
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Continuity of an integral with parameter-dependent domain of integration

I have an integral of the form: $$g(s)=\int_{D(s)}f(x)dx,$$ where $x\in\mathbb{R}^n$, $s\in\mathbb{R}$, $D(s)$ is a parameter-dependant subset of $\mathbb{R}^n$, $f$ is a "nice" function (i....
Carlos Santi Toledo's user avatar
2 votes
1 answer
225 views

Characterization of lower semicontinuity [duplicate]

I have a question related to lower semicontinuity of real valued function. Let $X$ be a metric space, $x_0\in X$ and $f:X\rightarrow\overline{\mathbb{R}}$ a function. We say that $f$ is lower ...
Raúl Filigrana Villalba's user avatar
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1 answer
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Inclusion of uppersemicontinuous maps in continuous maps

I have a question please. Is there a result about the inclusion of uppersemicontinuous maps in continuous maps ? I mean if F is uppersemicontinuous, is there always a map G continuous such that $F\...
Galois's user avatar
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Is this function upper-semi continuous?

Let $(A,\mathcal A)$ be a measure space, let $\mathcal P(A)=\{ q : (A, \mathcal A, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(A,\mathcal A)$. We equip $\mathcal P(A)$ with ...
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Counter example for: If $f\colon \mathbb{R}^n\to \mathbb{R}$ lower semi continuous, then $\forall x_n\to x, \lim_{n\to\infty}f(x_n)\geq f(x)$

We start with a definition. We say that a function $f\colon \mathbb{R}^n\to \mathbb{R}$ is lower semi continuous (l.s.c.) if $$\forall x\in\mathbb{R}^n,\; \liminf_{y\to x}f(y)\geq f(x).$$ As ...
Choripán Con Pebre's user avatar
3 votes
0 answers
56 views

$\lim_{k \rightarrow \infty} \frac{f(x)}{k}$ where $f: X \rightarrow \mathbb{R} \cup \{\infty\}$

I have this line of assumptions for some exercise to be proven "Let $X$ be a metric space and let $f: X \rightarrow \mathbb{R} \cup \{+\infty\}$ be a lower semi-continuous function which is ...
Iyad  Walweel's user avatar

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