Questions tagged [semicontinuous-functions]

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

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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 "...
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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 \...
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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 $c$, $\{x; \sum f_n(x)>c\}=\bigcup_{n=1}^\infty \{x; \sum_{k=1}^n f_k(x)>c\}$. ...
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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....
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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 ...
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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\...
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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 ...
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$\lim_{k \rightarrow \infty} \frac{f(x)}{k}$ 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 ...
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Lower semicontinuity of the rank for a map of vector bundles

In Nitsure's Cohomology of the moduli of parabolic vector bundles, we have the following Remark on page $62$: ([Remark 1.2) Let $E$ be a vector bundle on a scheme $S$ and let $F_1$ and $F_2$ be ...
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Measure of closed set is vaguely u.s.c. on M_1^+(K)

I'm working my way through Compact convex sets and boundary integrals by Alfsen and during the proof of Proposition I.2.8 they use that set $W=\{\mu\in M_1^+(K)|\mu(\overline{V})<\alpha\}$ is ...
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$f$ is lower semi-continuous at $x_0$ if and only if $\liminf f(x_k) \geq f(x_0)$ for all $x_k \rightarrow x_0$

Prove the following claim:$f$ is lower semi-continuous at $x_0$ if and only if $\liminf_{k\rightarrow \infty} f(x_k) \geq f(x_0)$ for all $x_k \rightarrow x_0$ as $k\rightarrow \infty$. First, assume ...
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Prove that $\Phi$ is a lower semicontinuous functional.

QUESTION: Let $(X, \|.\|)_X$ be a Banach space and $Y\subset X$ a subspace, which is itself a Banach space endowed with a norm $\|.\|_Y$ such that $\|y\|_X\leq \|x\|_Y$ for every $y\in Y$. Assume that ...
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Is the limit of an increasing sequence of continuous functions is a lower semicontinuous function [closed]

1.$ f_n(x)$ is continuous,bounded,positive and $f_n(x)\le f_{n+1}(x)$ 2.$ f(x)=\lim f_n(x)$ Question:Is $f(x)$ a lower semicontinuous function?
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Is there any function $f:\mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}$ such that $f$ is lower semi continuous only at $\mathbb{Q}$?

Is there any function $f:\mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}$ such that $f$ is lower semi continuous only at $\mathbb{Q}$ ? A function $f:\mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}$ is said ...
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Is this solution to part of Papa Rudin Chapter 2, Exercise 22 correct?

This is part of Papa Rudin Chapter 2 Exercise 22. Suppose that $X$ is a metric space, with metric $d$, and that $f:X \to [0, \infty]$ is lower semicontinuous, $f(p)< \infty$ for at least one $p \in ...
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Problem 3 chapter 12 Jost Postmodern analysis

Help solving this problem Let $g:\mathbb{R}\times(0,\infty)\rightarrow \mathbb{R}$ be defined by $g(x,y)=|xy-1|$, $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x):=\inf_{y>0}g(x,y)$. Show $f$ is NOT ...
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Upper bound for upper semicontinuous function in two dimensions

I have a right-continuous, upper semi-continuous operator defined by $$\mathcal{L}(\psi)(\overline{x}) := c_1 \psi_{x_1}(\overline{x}) + c_2 \psi_{x_2}(\overline{x}) - (\lambda+q) \psi(\overline{x}) + ...
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Adjoint for real functions

We want to find sufficient and necessary conditions for $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ to be adjoints with the usual order in $\mathbb{R}$, i.e $$f(x)\leq ...
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The infimum of a semi-continuous function on a compact set

Define a funtion as follows: $$f:X \rightarrow \mathbb{R}_+$$ with the property that $f(x)=0$ if and only if $x\in S$, where $\mathbb{R}_+=[0, \infty)$ and $X$ and $S$ are subsets of $\mathbb{R}^n$ ...
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A continuous, convex functional on a Banach space is weakly lower semicontinuous

Let $I:X \rightarrow \mathbb{R}$ be a continuous, convex functional on a Banach space $X$ (or Hilbert for instance). Then how to prove that $I$ is weakly lower semicontinuous. i.e $ \forall u_n \...
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Finite valued, semi-continuous functions on a measure space: is it locally constant almost everywhere?

Let $f:X\to [\![a,b ]\!]$ be a discrete-valued, lower-semicontinuous function on an arbitrary Borel measure space $X$. Let $A=\{x \in X: f \text{ is not locally constant at } x\}$ Question: does $A$ ...
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Checking lower semicontinuity for a function of two variables

Definition. Let $X$ be a finite vector space with inner product $\langle\cdot\vert\cdot\rangle_{X}$, and consider the extended real function $F: X\rightarrow \mathbb R_{\infty} := \mathbb R\cup \{\pm \...
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Lower semicontinuity of Lp integral with 0<p<1 with respect to L2 topology

Denote $I_p[u]:=\int_\Omega |u|^p$ with bounded $\Omega\subset\mathbb{R}^d$ and $0<p<1$. My question is: Is $I_p$ lower semicontinuous with respect to (strong) $L^2(\Omega)$-topology? I.e., does ...
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'Semi continuity' of function defined by inf\sup

Say I have a function $S:\Omega \to \mathbb{R}\cup \{ \pm \infty \}=:\mathbb{R}_{\pm \infty}$, and a set $X$ such that I match a subset $\mathcal{C}(x)\subseteq \Omega$ for all $x\in X$. I then define ...
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Convex Functions: Lower Semicontinuity and Epigraph

In the lecture notes by Christian Clason, on page 24, there is an interesting Theorem he proves (he calls it "Lemma 3.1"), namely: Let $F: X \to \overline{\mathbb R}$. Then $\text{epi} F$ ...
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Integral of a lower semicontinuous function is continuous

Let's say that a function $g$ is lower semicontinuous if the set $\{x:g(x)>\alpha\}$ is open for every real $\alpha$. Let $g:[a,\ b] \to \mathbb R$ be a lower semicontinuous function. Define $G$ by ...
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7 votes
1 answer
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Upper semicontinuity in real analysis

Exercise from book: Let $\left(f_{k}\right)_{k=1}^{\infty}$ be a sequence of functions and suppose that they are all upper semi-continuous at $x_{0}$. Define the function $g$ by $g(x)=\inf _{1 \leq k&...
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3 votes
1 answer
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Definition of Semicontinuity: Confusion in Rudin's RCA

Rudin's RCA defines upper and lower semicontinuity as follows: Let $f$ be a real or extended-real function on a topological space. If $$\{x: f(x) > \alpha\}$$ is open for every real $\alpha$, $f$ ...
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2 votes
1 answer
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Do I have the wrong interpretation of the limit inferior in this definition?

I am trying to understand the definition of lower semi-continuity. A function $f$ is lower semi-continuous at some point $x_0$ if the following holds $$\lim_{x \to x_0} \inf f(x) \geq f(x_0).$$ This ...
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2 votes
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How to prove the criteria for lower semicontinuity of a function at a point?

I have the following statement I need to prove. Let $f:D \to \mathbb{R}$. Then the function $f$ is lower semicontinuous at some point $x_0 \in D$ if $$\lim_{x \to x_0 } \inf f(x) \geq f(x_0)$$. I have ...
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A function $f$ is lower semicontinuous iff it is its own lower boundary.

The following is an exercise from Bruckner's Real Analysis: Prove that a function $f$ is lower semicontinuous if and only if it is its own lower boundary. Definition of semicontinuous : A function $...
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1 answer
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Why care about lower semicontinuous function?

I am a bit confused by certain area of math such as optimization is obsessed with lower semi-continuous (lsc) (and upper semicontinuous) functions, when continuity seems to describe all functions of ...
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On which spaces can lower semicontinuous functions be approximated from below by bounded continuous functions?

On metric spaces this can obviously be done by defining $F_n$ as in this question and then e.g setting $G_n(x) = \min \{ F_n ( x ), n \}$. But what if the space in question is not metrisable? In my ...
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Is this functional $\psi(u)=\int_{\Omega} \int_{0}^{u(x)} H(u(x)-\mu)dsdx$ is upper semi-continuous?

Let $X$ be a real Banach space, for $u \in X$ we define the following functional $$\psi(u)=\int_{\Omega} \int_{0}^{u(x)} H(u(x)-\mu)dsdx$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ...
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Does a function $f: E \rightarrow \mathbb R$ and such that $f$ is continuous, but not weakly lower semcontinuous exist?

Do an infinite Banach space $E$ and a function $f: E \rightarrow \mathbb R$ such that $f$ is continuous, but not weakly lower semcontinuous exist? I'm trying to find an example, but I can find only ...
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Weak lower semincontinuity of a functional with weak lower semicontinuity of $W^{1,2}$-norm

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Let $(f_n)_{n\in \mathbb{N}}\subset W^{1,2}(\Omega)$ be bounded. Then there is a subsequence $(f_{n_k})_{k\in \mathbb{N}}$ which converges weakly ...
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If $a(u,u) \leq \liminf a(u_n, u_n)$, is $a(u,u) \leq \liminf a(u_n, u_{n-1})$?

Let $a\colon H \times H \to \mathbb{R}$ be some bilinear form and $H$ is a Hilbert space. If it is weak lsc, i.e., $$a(u,u) \leq \liminf a(u_n, u_n)$$ for a sequence $u_n \rightharpoonup u$ in $H$, ...
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3 votes
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Lower Semi-Continuous Functions Arising from Plane Flows

If $X$ is a Hausdorff space, by a flow I mean a continuous surjection $F: X \times \mathbb{R} \rightarrow X$ such that $F(x, s + t) = F(F(x,s), t)$ for all $x \in X$ and $s, t \in \mathbb{R}$. If $x$ ...
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2 votes
1 answer
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Sum of lower semicontinuous functions

I'm a little confused about when the sum of two lower semicontinuous functions is continuous. I couldn't find a neat answer to my question on this site, though there are similar ones. Say that $f: \...
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When is the convex envelope lower semicontinuous for an infinite dimensional space?

Let $X$ be a infinite-dimensional Hausdorff real topological vector space and $f:X\to[-\infty, \infty]$ be a proper function. Define the convex envelope of $f$ to be the largest convex minorant of $f$....
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1 vote
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An approximation of a d-dimensional function using an order q<d function

Suppose we have a function $f:\mathbb{R}^d\to\mathbb{R}$. Using the notation of https://www.maths.unsw.edu.au/sites/default/files/amr08_5_0.pdf, which defines an order $q$ function $g:\mathbb{R}^d\to\...
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1 vote
1 answer
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Finiteness and Lower Semi-Continuity of an Functional.

Assume I have a probability density $\rho$ on $\mathbb{R}^n$ with finite second moment $$ \int_{\mathbb{R}^n}\|x\|^2\rho(x) dx<C. $$ I'm now interested in the following functional $$ F(\rho):=\int_{...
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How to prove the lower semi-continuity of this functionnal on $H^1(\Omega)$

I am studying the following application which goes from $H^1(\Omega)$ to $\mathbb{R}$ with $\Omega$ a bounded regular subset of $\mathbb{R}^3$ : $$H : u \mapsto \int_{\Omega} \left(|\nabla u|^2 - a \...
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0 votes
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A semicontinuous diagonal function?

Let $X$ be a compact Hausdorff space and let $(x_\alpha)_\alpha$ be a net in $X$ on the directed set $(A, >)$ that converges to $x$. Let $f: X \times X \to [0, \infty]$. Suppose $f$ has the ...
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Characterization of semicontinuity

Let $X$ be a metric space and $f:X\to\mathbb{R}$ be lower semicontinuous (LSC) at $x\in X$, i.e. $\liminf_{y\to x} f(y)\geq f(x)$. A function is called LSC if it is pointwise LSC. For my thesis I have ...
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Semi continuity is continuity with respect to one sided order topology?

I've noticed that semi continuity is a 'sort of' continuity with respect to one sided limits: We say that $f:X\to \mathbb{R}$ is upper (or respc. lower) semi-continuous at $x_0\in X$ if for all $\...
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1 vote
1 answer
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Converse to extreme value theorem?

It's well known that continuous functions achieve minima on compact sets. An even stronger result is that lower semicontinuous functions achieve minima on compact sets. Question. If an extended-real ...
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When continuity implies weak continuity

Let $X$ be a separable Hilbert space. Let $f:X\rightarrow \mathbb{R}$ be (lower semi-) continuous with respect to the weak topology. Then, I have read that if in addition, $f$ is convex, then it is ...
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4 votes
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A concave function on a compact and convex set is upper semi continuous

Let $X$ be a compact and convex set and $f:X \to \mathbb{R}$ be a concave function. Therefore, $f$ is continuous on interior of $X$. I want to know whether $f$ is upper semi continuous on $X$. If not, ...
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