Questions tagged [semicontinuous-functions]

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

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Strong lower semicontinuity theorem

In the book Dacorogna - "Direct methods in the calculus of variations", corollary 3.22 on page 94 states: Let $p \geq 1$, $\Omega \subseteq \mathbb R^n$ open and $f:\Omega \times \mathbb R^{...
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17 views

Lower limit and subsequence

Let $x_n$ be a convergent sequence $x_n \to x$ in a topological space $X$ and $F:X\longrightarrow \mathbb R$ any function (not necessarily continuous). If there exist a subsequence $x_{n_k}$ such that ...
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13 views

Regarding ln|f| being upper semicontinuous

Let $f$ be a holomorphic function on the open unit disc $\mathbb{D}$ In $\mathbb{C}$. Can anyone tell why $\ln|f|$ is upper semicontinuous but not continuous? In particular $\ln|z|$, $z\in \mathbb{D}...
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26 views

Lower semi-continuous on compact set

Definition of lower semi-continuous: Give topo space X and mapping $f:X \to \left(-\infty,+\infty\right]$. $f$ is lower semi-continuous at $x_0$ if $\forall \varepsilon > f(x_0)$, $\exists V$ is ...
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Lower Semicontinuity Equivalence

Let be $(X , ||\cdot||)$ a normed space and $f:X \longrightarrow \mathbb{R}$ a function. $f$ is lower semicontinuous if $\{x \in X:f(x) \leq c \}$ is closed $\forall c \in \mathbb{R}$, and is said to ...
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59 views

Is the functional $I(u) = \int_{\Bbb{R}^N}h(x) |u|^q \ dx $ weakly lower semicontinuous?

Is the functional $$ I(u) = \int_{\Bbb{R}^N}h(x) |u|^q \ dx $$ weakly lower semicontinuous for $N \geq 3$ and $1 < q < 2$? This is part of an exercise that asks to solve the problem $$ \begin{...
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72 views

Proving that a function is weakly lower semi continuous

Let $X$ be a normed vector space and $g : \mathbb{R}^+\to \mathbb{R}$ a monotonic increase function that is lower semi continuous. Then, I want to show that $h := g(\|\cdot\|)$ is weak lower semi ...
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3answers
61 views

References on semicontinuous functions

The generalisation of continuity to semicontinuity is well-known. I suppose it should be also well-studied. The only references I found offhand are the ones from the wikipedia entry semi-continuity. ...
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20 views

Prove that a simple function is lower semi-continuous at a point

Using the definition that a function $f : A \to \mathbb{R}\cup\left\{ -\infty, \infty \right\}$ is l.s.c at a point $\bar{x} \in A$ if $\forall \alpha < f( \bar{x} ), \exists \delta > 0$ s.t. $\...
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Is it required that singletons of a finite decomposition be closed in order for it to be upper semicontinuous?

In Willard's "General Topology", exercise 9E (p67 of the Dover edition), he writes: A decomposition $\mathscr{D}$ of a space $X$ will be called finite iff only finitely many elements of $\mathscr{D}...
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Optimal Stopping - Reference Request

I am considering an optimal stopping problem of the form: $$v(x,t)=\sup_{\tau}\mathbb{E}_{x}\left[g(x_{\tau})+a(t+\tau)\right]$$ This problem, with some requirements on $g$ and $x$, can be put into a ...
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29 views

Prove that $s$ is constant $\mu$-almost everywhere if $s$ is upper semicontinuous

Let $f:(X, d)\to (X, d)$ be a homeomorphism on compact metric space $(X, d)$. We define a new metric $d_\infty$ on $X$ by $d_\infty(x, y)= \sup_{n\in\mathbb{Z}}d(f^n(x), f^n(y))$, and $s(x)=\inf \{d_\...
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Weak lower semicontinuity of functional with two arguments

Let $\Omega$ be a bounded and smooth domain and let $J:H^1(\Omega) \times H^1_0(\Omega) \to \mathbb{R}$ be defined by $$J(u,v) = \int_\Omega f(u)|\nabla v|^2$$ where $f\colon \mathbb{R} \to \mathbb{R}...
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1answer
30 views

Upper semicontinuous function on a tangent bundle

Let $\Omega$$\subset\mathbb{C}^n$ be a open connected set (domain). Let $T(\Omega)$ denote the tangent bundle of $\Omega$. Let $f:T(\Omega)\longrightarrow [0,\infty)$ be a map on $T(\Omega)$. What ...
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Question about semi-continuous functional

My research is dealing with a functional $f(x,y): \mathbb{R}^n \times \mathbb{R}^k \to \mathbb{R}^+$, that I know is lower-semicontinuous in both $x$ and $y$. I would like the following to be true, ...
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124 views

Lower Semicontinuous Function = Supremum of Sequence of Continuous Functions

Background I'm reading Cedric Villani's Optimal Transport: Old and New [1] and came across a result (below) I'm not quite sure how to prove. It is used to prove Lemma 4.3 and through my research, I've ...
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36 views

$\mathbb{1}_{\mathbb{R}\setminus \mathbb{Q} }$ as infimum of a sequence of lower semi-continuous functions

I have been trying to construct a sequence of lower semi-continuous functions that will give indicator function as the infimum. In particular, I need a sequence $(f_i)_{i\in I}$ of lsc functions such ...
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1answer
25 views

Upper semi continuous equivalence

I have a doubt about upper semicontinuous functions. I have two definitions and I need to show that they are equivalente. $f$ is upper semi-continuous at $x_{0}$ if for every $\varepsilon>0$, ...
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uppersemicontinuous correspondence, fixed point and compact set

I want to know a counter example which opposes 'Every upper-hemicontinuous correspondence with nonempty and compact values defined on $[0,1]^2$ has a fixed point". Can somebody gives me one? By the ...
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57 views

Why is vitali caratheodory theorem important?

The Vitali Caratheodory theorem, as stated in Rudin's real and complex analysis, states that for nice measures $\mu$ we can approximate any $L^{1}$ function $f$ by upper lower semicontinuous $u$ and ...
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Approximate lower semi-continuous functions with uniformly continuous functions [duplicate]

If $(X,d)$ is a metric space and $F$ is a nonnegative lower semi-continuous function on $X$, then $F$ can be written as the sup of increasing sequence of uniformly continuous functions: $F_n(x) = \...
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Show that $\exists \rho\in (a,b)$ such that $f(\rho)\le f(x)$ forall $ x\in (a,b)$.

I am stuck on the following problem: Let $f:(a,b)\to \Bbb R$ be lower semicontinuous. Also $\lim_{x\to a+} f(x)=\lim_{x\to b-} f(x)=\infty$. Show that $\exists \rho\in (a,b)$ such that $f(\...
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64 views

Equality of definitions of Lower Semicontinuity [closed]

I would like to show the equality (i.e. iff) of the following definitions of lower semicontinuity without the use of liminf. Definition: Lower Semicontinuous Let $(X, \|\cdot\|)$ be a normed vector ...
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41 views

semi-continuous function

Given $f: \mathbb{R} \to \mathbb{R}$ and \begin{align*} f(x) = \begin{cases} x^2 & \text{if}\; x \not= 0 \\ -1 &\text{if}\; x = 0 \end{cases} \end{align*} Please help me prove it follows ...
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Let $f$ and $g$ have compact sublevels. Does it imply that $f + g$ is lower semicontinuous?

Let $(X, d)$ be a metric space, and $f, g : X \rightarrow \mathbb{R}\cup\{\infty\}$ be proper. Let further $f$ and $g$ have compact sublevels. Does it imply that $f + g$ is lower semicontinuous (lsc)?...
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62 views

Integral of a lsc function is lsc?

Let $f:\mathbb{R}^d\times \mathbb{R}^D\rightarrow \mathbb{R}$ be a lower semicontinuous function. Then is it true that $$ \begin{aligned} L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)&\...
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76 views

Continuous function between a lower semi-continuous function and an upper semi-continuous function.

Let $X$ be a compact metric space, $u: X \to [0, 1]$ an upper semi-continuous function and $l: X \to [0, 1]$ a lower semi-continuous function such that $u(x) < l(x)$ for each $x \in X$. Does ...
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Which topology is transferred by this specific metric on the rectangle $[0,\infty] \times [0 , \infty ]$?

Let $\varphi : [0,\infty]^2 \to [0,1]^2$, $(t,x) \mapsto (\frac{t}{1+t} , \frac{x}{1+x})$. The inverse is given by $\varphi^{-1}: [0,1]^2 \to [0,\infty]^2$, $(t,x) \mapsto (\frac{s}{1-s} , \frac{x}{1-...
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1answer
77 views

Equivalent definitions of semicontinuity

I am trying to understand the equivalence between two definitions of an upper semicontinuous real-valued function over a compact normed space $X$. Suppose that for every $y$ in some set $Y$, the ...
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1answer
58 views

If $f$ is LSC, then there exist continuous $g_1\le g_2 \le \dots\to f$ pointwise everywhere.

This is Exercise 2.22 from Rudin's Real and Complex Analysis. I can't prove this in the case where $f$ may take infinite values. Suppose that $X$ is a metric space with metric $d$, and that $f:X \to [...
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1answer
44 views

Prove that a function is sequentially lower semicontinuous

Let be $(X, \{ p_i \}_{i \in I} )$ a locally convex space, $M_0\subset X$ a bounded and nonempty set and $f = l + I_{M_0}$ where l is a continuous function and \begin{equation*} I_{M_0}(x)= \begin{...
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1answer
149 views

Conjugate Function Is Lower Semicontinuous

Let $E$ be a Normed Vector Space. Let $\phi : E \rightarrow ( - \infty , + \infty] $ be a function such that $\phi$ is not equivalent to $\infty$. i.e. The set $ \{ x \in E : \phi (x) \neq \infty \}$ ...
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56 views

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 $f\square g: x\mapsto \inf\{f(y)+g(x-y): y \in E\}$. Here I do not assume any ...
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79 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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167 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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27 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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65 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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50 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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521 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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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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144 views

Proximal Gradient Method Justification - Why Does It Work?

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
60 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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1answer
191 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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108 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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1answer
106 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 two ...
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1answer
102 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
80 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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1answer
88 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
56 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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1answer
310 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 ...