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Questions tagged [semicontinuous-functions]

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

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Is the inf-convolution of two continuous functions continuous?

For two continuous functions $f$ and $g$ defined on a normed space $E$ taking values in $[-\infty,\infty)$, let $f\square g: x\mapsto \inf\{f(x)+g(x-y): y \in E\}$. Here I do not assume any ...
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43 views

Reference for extreme value theorem.

I look for a reference of the Extreme Value Theorem for semicontinuous functions defined on a topological space. I know the proof, but I want to cite this result in my work.
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82 views

Why the length is lower semi-continuous?

I'm reading the book A Course in Metric Geometry by Dmitri Burago, Yuri Burago Sergei Ivanov and I don't understand the proof of Proposition 2.5.17 (Page 48). More precisely, I don't know why the ...
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1answer
20 views

An inequality about compact functions and lsc functions

I'm reading a proof and it contains the following inequality: Suppose $u: E \rightarrow [0,\infty]$ is lower semicontinuous and let $u_t$ be a sequence of Lipschitz functions approaching $u$ from ...
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63 views

How far from a matrix the rank does not decrease?

Let $A$ be a real $n \times n$ matrix, and suppose that $\text{rank}(A)=k$. Consider $$ r_0=\sup \{r>0 \,\, | \,\, |B-A| \le r \Rightarrow \text{rank}(B) \ge k \}.$$ Does $r_0$ depend only on ...
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36 views

Silly question on lower semi-continuity

Suppose that $X_n\rightarrow X$ in a complete separable metric space $(\mathcal{X},d)$. Let $f:\mathcal{X}\rightarrow (-\infty,\infty]$ be a proper, convex, lower semi-continuous function, such that $...
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2answers
74 views

Show that lower semi-continuous function attains it's minimum. (Proof verification) (By contradiction)

Let $f: [0,1]\to \mathbb{R}$ be a lower semi-continuous function, then $$ \liminf_{x\to a} f(x) \geq f(a), \forall a \in [0,1]$$ I have to prove that $f$ attains its minimum on $[0,1]$, that is: $\...
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45 views

Do separately semi-continuous functions have a dense set of semi-continuities?

The connection between separate continuity and joined continuity has been studied quite a lot. In particular, one has (as a special case of a far more general Theorem from here) the following: If $...
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2answers
64 views

Proximal gradient method justification

If $f$ and $g$ are respectively a differentiable function and a convex, lower semi-continuous function, then the algorithm defined by: $$ x^{k+1} = \text{prox}_{\gamma{g}}[x^{k} - \gamma\nabla{f(x^{k}...
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1answer
38 views

Upper semicontinuous function as a poinwise limit of continuous fuctions

The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous ...
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55 views

Lower semi continuous envelope is lower semi continuous

Let X be a topological space and $F:X \rightarrow \overline{\mathbb{R}}$. The lower semi continuous envelope of $F$ is defined by $sc^-F(u)=\sup\{\phi(u)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} ...
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44 views

Functional is weak lower semicontinuous but not weak continuous

I want to show that the functional $L(u)=\int_0^1 \sqrt{1+(u'(x))^2} dx$ is lower semicontinuous in terms of weak convergence in $W^{1,p}(0,1), p\in(1,\infty)$ but not continuous. Our definition of ...
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68 views

Show that support function of any set in $\mathbb{R}^n$ is lower semi-continuous function.

Let $A \subseteq \mathbb{R}^n$. The support function of set $A$ is defined as the following $$ S_A(x)=\sup_{y \in A} x^Ty $$ where $x \in \mathbb{R}^n$. To show it is lower semi-continuous we have ...
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45 views

Upper semicontinuous functions to $\mathbb{N}$ are locally constant on a dense subset

Let us take $f : X \rightarrow \mathbb{N}$ an upper semicontinuous function. In Wikipedia - Semi-continuity it is said that such a function must be locally constant on a dense open subset. I don't ...
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1answer
29 views

Show a function mapping a metric space to $\mathbb{R}$ is continuous if and only if it is both upper and lower semi-continuous

Let $(M,d)$ be a metric space and $f:(M,d)\rightarrow \mathbb{R}$. Show that $f$ is a continuous function if and only if it is both upper and lower semi-continuous. Definition: A function is lower ...
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32 views

a proof that the pointswise limits of lower semicontinuous (lsc) functions is lsc

I have a question regarding a proof regarding lower semicontinuous functions (the proof of the claim below). Definition: We call a function $f\colon \mathbb{R}^n\to\mathbb{R}\cup \{\infty\}$ lower ...
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1answer
48 views

Convex sets and semicontinuous functions

I have this problem: Let $X$ be an open bounded subset of $\mathbb{R}^n,$ and fix $x_0\in X.$ For all $e\in S^{n-1}$ we put $$\phi(e)=\sup\{t\geq0: x_0+te\in X\}$$ $$\overline{\phi}(e)=\inf\{t\geq0: ...
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136 views

Show that the function $h$ defined by $h(x)=g(f(x))$ is lower semicontinuous.

Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ be a lower semicontinuous function, and $g:\mathbb{R}\longrightarrow\mathbb{R}$ be a lower semicontinuous and nondecreasing function. (1) Show that the ...
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65 views

prove a function is lower semicontinuous

$E = \ell^p$, with $1\le p<\infty$. For $x\in\ell^p$, $x = (x_1,x_2,\dots,x_n,\dots)$, check function $$\varphi(x) = \begin{cases}\sum_{k=1}^{\infty}k|x_k|^2 &\text{ if } \sum_{k=1}^{\infty}k|...
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51 views

One-Sided Notion of Topological Closure

Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this: Let $A$ be a subspace of $\mathbb{R}$. Define an operation ...
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95 views

Show that $\theta(x)$ is upper semi-continuous [closed]

Let $\mu$ be a Borel measure on $\mathbb{R}^n$. Let $\rho >0$, a fixed positive number, and $B_{\rho}(x)=\{y\in \mathbb{R}^n:d(x,y)<\rho\}$. For $x\in \mathbb{R}^n$, define a function: $$\theta(...
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1answer
49 views

Subharmonic on $U$ iff subharmonic on each $U_{\alpha}$, where $(U_{\alpha})$ is an open cover of $U$

I am working with this definition of subharmonicity: Definition Let $U$ be an open subset of $\mathbb{C}$. A function $u: \, U \to [-\infty,\infty)$ is called subharmonic if it is upper ...
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50 views

How to show lower semicontinuity: differentiability $\rightarrow$ continuity $\rightarrow$ lower semicontinuity?

Take $G: \mathbb{R}^M\rightarrow \mathbb{R}$ with $G(a)\equiv \mathbb{E}_{\mathbb{P}}(\max_{k\in \{1,...,M\}}V_k+a_k)$ for any $a\equiv (a_1,...,a_M)\in \mathbb{R}^M$, where: A1: $V\equiv(V_1,...,...
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101 views

Points of discontinuity of a semicontinuous function on a complete metric space.

I was studying the book 'Recurrence in Ergodic Theory and Combinatorial Number Theory' by Harry Furstenberg. In Chapter 1, the last lemma which he proved is : Lemma 1.28$\:$: ...
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51 views

$f$ is LSC at $x$ if and only if $\lim_{\delta \to 0}\inf\{f(y) | y \in B(x,\delta)\}=f(x)$

Asuume $f$ is LSC(lower semicontinuous) at $x$ . If $t\lt f(x)$, then there exsits $\delta \gt 0$ such that $t\lt f(y)$ for all $y \in B(x,\delta)$. Thus, $t \le \inf\{f(y)| y\in B(x,\delta)\}$ and we ...
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37 views

For a continuous function defined on [a,b] , is the set of points at which f(x)>d closed set? [closed]

Prove that if f(x) is a continuous function defined on [a,b] , then the set of points at which $f(x) \geq d$ is closed, for all numbers d.
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How to remember which is lower/upper semicontinuity?

There are several ways in which continuity can be formulated as two conditions - in a way such that one of them is lower semicontinuity and the other one is upper semicontinuity. (See below for ...
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30 views

Lower semicontinuous submeasure is countably subadditive

A submeasure is a function $\phi\colon\mathcal P(\mathbb N)\to[0,+\infty]$ that is monotone and subadditive. If, additionally, $$\phi(A) = \lim\limits_{n\to\infty} \phi(A\cap[1,n])$$ then it is called ...
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1answer
35 views

The reason for a certain requirement in upper-semicontinuity

Let $(\Bbb{X}, \Sigma, \mu)$ be a measure space, The property of upper semicontinuity means that if $A_n$ is a decreasing sequence of measurable sets, namely $A_1 \supset A_2 \supset \ ...$ , and $\mu(...
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91 views

The seminorms that give the strict topology on the space of bounded continuous functions

If $X$ is a completely regular space, then the strict topology on the algebra of its bounded continuous functions is usually taken to be given by the seminorms $f \mapsto \| f \|_\varphi = \sup \...
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1answer
75 views

“Upper derivative” of indefinite integral of upper semicontinuous function

The following problem is stated as Exercise 22.A(iii) in the book Van Rooij, Schikhof: A Second Course on Real Functions. Let $f\colon [a,b]\to{\mathbb R}$ be Lebesgue integrable and upper ...
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81 views

Baire space upper semicontinuos map

In the metric space $(Z,d)$, let $A(z_0,\varepsilon)$ denote the closed ball $\left\lbrace z\mid d(z,z_0)\leq\varepsilon\right\rbrace$. Now let $X$ be an arbitrary space, let $Y$ be a metric space and ...
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305 views

Show that the supremum of a collection of lower semicontinuous function is lower semicontinuous

I know there's already a question with a title very similar to this, unfortunately as I understand the OP skips over the part of the proof that is not clear to me. Let $I$ be a set and $f_\alpha$, $\...
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1answer
169 views

Uniform sum of positive upper semicontinuous functions is upper semicontinuous?

Let $X$ be a metric space. A real-valued function $f : X \rightarrow \mathbb{R}$ is upper semicontinuous if it satisfies one of the followings: $(1)$ For all $c \in \mathbb{R}$, its preimage $f^{-1}(...
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1answer
138 views

Prove that $Uf = f$ if and only if $f$ is upper semicontinuous

Let $X$ be a metric space. $f:X \rightarrow \mathbb{R}$ is upper semicontinuous if for all $\varepsilon>0$ and all $x \in X,$ there exists an open neighbourhood $U \ni x$ such that for any $y \in U,...
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484 views

Prove by definition that every upper semi-continuous function can be expressed as infimum of a sequence of continuous functions.

Suppose $X$ is a metrizable space. A real-valued function $f:X \rightarrow \mathbb{R}$ is upper semicontinous if for any real number $c$, its preimage $f^{-1}(-\infty,c)$ is open in $X$. In this post,...
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1answer
171 views

Lower-semicontinuous submeasure on $\mathbb N$ vs. function on Cantor space

On the page 7 of Farah's Analytic Quotients, immediately after the definition of lower semicontinuous submeasure (see below) there is a brief remark saying that "This obviously corresponds to $\phi$ ...
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349 views

Upper-semicontinuous functions and the condition $\limsup_{x\to p} f(x) \le f(p)$

When looking at Wikipedia article about semi-continuity I saw equivalent condition for upper-semicontinuity using limit superior. However, I think that the condition that $X$ is metric is redundant ...
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448 views

Uniform limit of upper semi-continuous functions

I want to show that if $\{ f_n \}_{n\in\mathbb{N}}$ is a sequence of upper semi-continuous functions defined, for example, on a set $G\subset\mathbb{R}$ which converges uniformly on $G$ to a function $...
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85 views

For bounded real valued function $f$ show that $\omega_f$ is upper continuous

Let $f:X\to\Bbb R$ be bounded, where $X$ is a metric space. Let the function $$\omega_f(x):=\inf_\epsilon \omega_f(x,\epsilon)=\lim_{\epsilon\to 0^+}\omega_f(x,\epsilon)\tag{1}$$ where $$\...
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1answer
190 views

Discontinuities of upper semicontinuous function

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an upper semicontinuous function (USC), i.e. for all $\alpha \in \mathbb{R}$, the set $\{ x \in \mathbb{R} : f(x) < \alpha \}$ is open. What are ...
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1answer
158 views

Prove ceiling function is lower semicontinuous

Can someone give me a hand with this exercise please? I want to prove that the ceiling function is lower semicontinuous, but I am not sure how to do it. It comes in my book as an example, right after ...
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189 views

Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function?

[Q.] Is there a semicontinuous function, which has its discontinuous set with non-zero measure? Remark: Given a semicontinuous function, the set of all discontinuous points may be uncountable, for ...
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350 views

Lower semicontinuity of length of graph: $L(g)\le\liminf_{n\to\infty}L(f_n)$

I am interested in the following claim:$\newcommand{\intrv}[2]{[#1,#2]}\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}}$ Let $g,f_1,f_2,\dots$ be continuous functions on $\intrv ab$ such that $g=\...
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1answer
2k views

To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous

Show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous. Suppose $\{f_n\}$ is a sequence of lower semicontinuous functions on a topological space $X$. ...
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1answer
981 views

Composition of lower semicontinuous function with continuous function is lower semicontinuous

Assume that $f\colon \mathbb{R}^n \to\mathbb{R}$ is lower semicontinuous at $g(a)$ and $g\colon \mathbb{R}^m \to\mathbb{R}^n$ is continuous at $a \in\mathbb{R}$. Define $h = f \circ g \colon \mathbb{...
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1answer
684 views

Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions. My attempt: I don't know how to approximate $f(x)$ to ...
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2answers
2k views

What is the intuition for semi-continuous functions?

Here is the definition of semi-continuous functions that I know. Let $X$ be a topological space and let $f$ be a function from $X$ into $R$. (1) $f$ is lower semi-continuous if $\forall \alpha\in R$,...
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1answer
131 views

Is $x\mapsto \|Tx\|$ lower semi-continuous?

Suppose $T:\mathcal D(T)\rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$. Is it true that $$ \|Tx\|\leq \liminf_{n\rightarrow\infty} \|T x_n\...
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2answers
334 views

Continuity set of a difference of two upper semi-continuous real functions over a metric space

The difference of two upper semi-continuous functions is in general neither upper- nor lower semi-continuous. But what can be said about the continuity set of such a functions, specifically its ...