Questions tagged [semicontinuous-functions]

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

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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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16 views

Riemann integrability of upper semicontinuous functions [closed]

Is every upper semicontinuous function on a compact interval is Riemann integrable?
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22 views

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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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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50 views

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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37 views

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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18 views

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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56 views

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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71 views

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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20 views

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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27 views

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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25 views

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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23 views

Equivalent property for lower semicontinuous functions

In a book that i am currently reading i came across the following statement. Let $E$ be a polish space with a complete metric $d$ and $f : E \rightarrow [-\infty , \infty ] $ a lower semicontinuos ...
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17 views

Where am I going wrong with using the limit suprema definition of upper-semicontinuity for this particular function?

Suppose you have the following function (excuse the drawing): some lower-semicontinuous function This function is not upper-semicontinuous at $x_0$. I would like to verify this using the following ...
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50 views

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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19 views

$\psi$ is convex and lower semicontinuous $\iff$ $\exists \zeta$, $\psi(x)=\sup\limits_y(\zeta(y)+x\cdot y)$

I was trying to prove that a function $\psi:\mathbb{R}\to\mathbb{R}\cup\{\infty\}$ is convex and lower semi-continuous if and only if there exists a function $\zeta : \mathbb{R}\to\mathbb{R}\cup\{\pm\...
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8 views

Semicontinuous maps on Banach lattices

Let X be a Banach lattice in the partial order $\preceq$ and $K\subseteq X$ be a complete sublattice. Can we define upper semicontinuous maps from K into K as follows? A map $f:K\to K$ is said to be ...
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24 views

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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88 views

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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96 views

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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24 views

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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10 views

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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1answer
41 views

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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1answer
66 views

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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27 views

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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36 views

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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40 views

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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57 views

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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34 views

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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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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58 views

Two semi-continuous functions whose sum is nowhere semi-continuous

Prove or disprove the following statement: There exist functions $f,g:\mathbb{R}→\mathbb{R}$ such that: (1) $f,g$ are semi-continuous, i.e. for each $x\in\mathbb{R}$, $f$ is either upper semi-...
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94 views

If $u^*(x) = \lim_{r \to 0} \sup_{B_r(x)} u$ then $u^*$ is the smallest upper semicontinuous funtion which is greater then $u$

In the context of vicosity solutions for fully nonlinear PDE of second order (I don't know first order theory) it is useful to define the upper semicontinuous envelope of $u : \Omega \longrightarrow \...
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40 views

Proof of semicontinuity

Given Definition: Function $f$ on metric space $M$ is called lower semicontinuous if, for each real $a$ the set $\{x \in M: f(x) \leq a\}$ is closed in $M$. Question: Prove that $f$ is lower ...
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45 views

Is $\mathbb R$ completely normal space?

Is $\mathbb R$ completely normal ( $T_5$ ) space? I've seen this Wiki Page about Urysohn's lemma, which states that a topological space is normal if and only if any two disjoint closed sets can be ...
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31 views

Why do we need to invoke the discrete-topology in the following definition of right-continuity?

In a book I read the following sentence: "Let $V$ be a countable set, equipped with the discrete topology. A function $f : [0, \infty) \rightarrow V$ is right-continuous if $\lim_{s \downarrow t} ...
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32 views

Is there any redundancy here?

Let $\Phi$ be the class of all functions $\phi:[1, \infty) \rightarrow[0, \infty)$ satisfying the following properties: $\phi$ is lower semi-continuous; $\phi(1)=0$; for each sequence $\left\{t_{n}\...
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34 views

Examining an upper semi-continuous function on the empty topology

Let me start out with the question that prompts this: Original Problem: Let $X$ be a compact topological space, and let $f : X \to \Bbb R$ be an upper semi-continuous function. Show that $f$ attains ...
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121 views

semi continuity

I've read about semi continuity on set valued-maps from topological space and real functions. Given $X$ and $Y$ are two topological spaces and set valued function $$F:X\to \mathcal{P}(Y).$$ Function $...
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58 views

Question on showing function is lower semicontinuous

A function is lower semicontinuous at $x \in\mathbb{R}$ if given any sequence $x_n$ converging to $x$, $$f(x)\leq\lim_{n \to \infty} f(x_n)$$ Provided that the limit on the right exists. Let $$I_{u}(x)...
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Sufficiency of upper semi-continuity of $f$ and lower semi-continuity of the functions $f_n$ in Dini's theorem

I am referring to the formal statement and the proof of Dini's theorem as given on Wikipedia(the functions $f_n$ are increasing). It is sufficient that the functions $g_n$ are upper semi-continuous ...
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18 views

Is there a mistake in this passage about lower semi-continuity?

Im reading a paper, where I find this problem: Let $f, g$ two function such that from a Banach space $\mathbb R$ into itself: $f$ is continuous, and $g$ is lower semi-continuous; $(x_n)_{n\in \...
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39 views

Function series of normal lower semi-continuous functions

For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is defined as follows: $ f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in U\right\} :U\in ...
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101 views

Coercive/(weakly) semicontinuous function: extreme values

Consider functionals of the form $$\phi : X \rightarrow \mathbb{R} \cup\{+\infty\},$$ where $X$ is an arbitrary, normed vector space. In particular, $X$ may be of infinite dimension. I would be fine ...
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26 views

Extending the domain of a lower semicontinuous function

Let $D$ be a closed nonempty set in $\mathbb{R}^{m}$. Let $f: \mathbb{R}^{m} \times (0, \infty) \to \overline{\mathbb{R}}$ be a lower semicontinuous function such that $f(u, r) \nearrow \delta_{D}(u)$ ...
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39 views

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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22 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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19 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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63 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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175 views

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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75 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{...