# Questions tagged [semicontinuity]

The semicontinuity tag has no usage guidance.

42
questions

3
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### Semicontinuity of Exts

Let $Z$ be a proper noetherian scheme over an algebraically closed field $k$, and $S$ a smooth scheme over $k$. Let $\cal E$, $\cal F$ be coherent sheaves on $Z\times S$, flat over $S$. Is it true ...

0
votes

1
answer

132
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### Right-continuity of covering number

Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...

2
votes

1
answer

92
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### A question about a realcompact space and upper semicontinuous function

Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of
John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative
upper ...

2
votes

2
answers

438
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### A net of lower semicontinuous functions

Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise.
Please is it true that one can extract a countable ...

2
votes

0
answers

59
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### Conditions ensuring that a compact set can be approximated 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
$$
d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0.
$$
...

0
votes

0
answers

82
views

### 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 $x\in K$, ...

2
votes

1
answer

244
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### Semi-continuity in quasi-finite morphisms without properness

Let $f:U \to V$ be a flat, quasi-finite, surjective morphism between two affine varieties defined over $\mathbb{C}$. Assume that every closed fiber is reduced. Consider the function $\eta$ that sends ...

1
vote

1
answer

50
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### Weak lower semicontinuity of a sequence of Riemann sums

Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums
$$R^K=\frac{1}{K} \sum_{k=0}^{...

2
votes

1
answer

315
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### Is a convex, lower semicontinuous function that is bounded from below, actually continuous?

While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space.
Lemma: Let $f ...

2
votes

3
answers

254
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### Looking for a reference: $f$-divergences are lower semicontinuous

I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability ...

0
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0
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251
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### Flatness of affine cone due to semicontinuity theorem

I would like to clarify an important aspect from the discussion in this question.
The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic
Geometry Chap. III page ...

1
vote

0
answers

58
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### Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?

Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...

0
votes

1
answer

103
views

### Lower semi-continuity of induced function on sequences

Let $f:X\rightarrow [0,\infty)$ be (resp. weakly) lower semi-continuous on the reflexive Banach space $X$. Let $\ell^p(X)$ denote the space of $p$-summable sequences in $X$, i.e.: $\sum_{n=1}^{\infty}...

2
votes

0
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98
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### Semicontinuity of length for coherent sheaves

Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \...

2
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0
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### Weak relaxation of a strongly lower semi-continuous functional

Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...

2
votes

1
answer

268
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### 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 ...

2
votes

1
answer

109
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### Lower semi-continuity of length-dependent functional

Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional
$$
\begin{aligned}
F_f:&\ell^1 \rightarrow [0,\infty]\\
(x_n)_{n=0}^{\infty} &\to \sum_{n=...

1
vote

2
answers

104
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### Rank of a linear combination of linear operators

I asked this question a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow.
Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now ...

2
votes

1
answer

145
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### Regarding upper semicontinuity of a function

Let $E$ be a linear subspace of $\mathbb{C}^{n\times n}$.
Define the function $\mu_E:\mathbb{C}^{n\times n}\longrightarrow \mathbb{R}_+$ as
$$
\mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-...

2
votes

1
answer

92
views

### Weak lower semicontinuity of functional with two arguments

Let $\Omega$ be a bounded domain (smooth if necessary) 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} \...

0
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0
answers

98
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### Help showing F is weakly lower semicontinuous

Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...

0
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2
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282
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### Goldowsky-Tonelli theorem for upper semi continuous function

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. ...

5
votes

1
answer

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### When do convexity and lower semicontinuity imply continuity?

Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous.
Question: ...

3
votes

0
answers

414
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### Lower semicontinuous and convex envelope

L.Ambrosio, in paper [1] writes:
Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)
for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of ...

0
votes

1
answer

88
views

### If a function follows another one's range order, can we say it follows some continuity properties?

Suppose that $(X,d)$ be a complete metric space, $f:X \to \mathbb{R}$ is a sequentially lower monotone function. Let $g: X \to \mathbb{R}$ be a function with the property: $f(x)\leq f(y) \Rightarrow g(...

2
votes

1
answer

132
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### Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...

0
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1
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223
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### An extension for lower semi continuous lower bounded real valued functions class

Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...

4
votes

1
answer

211
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### Continuity concepts for correspondences

Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F ...

2
votes

0
answers

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### lower semicontinuity of the number of extreme points

Do you know the reference for the following fact:
the number of extreme points of a compact convex
subset of a locally compact space varies lower semicontinuously when we endow the space of compact ...

1
vote

1
answer

142
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### Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?

Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...

2
votes

0
answers

399
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### When will the upper regularization of a bounded function not defined?

Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...

1
vote

1
answer

942
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### properties of orderd upper and lower semi continuous functions [closed]

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.
If $f(x_0) = g(x_0) $ for some point $x_0\in M$,
is it ...

1
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0
answers

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### Semi-continuity extends to dual

Say $f \colon X \to \mathbb{R}$ is a lower semi-continuous on a compact space $X$. Let $\mathcal{P}(X)$ denote the space of Borel probability measures on $X$, and let $f^* \colon \mathcal{P}(X) \to \...

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votes

2
answers

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### Upper semicontinuity of set-valued maps with open values

Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as:
Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at ...

2
votes

1
answer

111
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### Lebesgue measurability of singular set

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and $f:Q\to\mathbb{R}$ be continuous function.
Define a superdifferential of $f$ at $x\in Q$ by
$$
D^{+}f(x)=\{p\in\mathbb{R}^{d} \mid \text{$f(y)\...

1
vote

0
answers

857
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### Approximation of semicontinuous functions by continuous (or smooth) functions with closed form

I'm looking for a sequence $(f_{\epsilon})_{\epsilon>0}$ of continuous (or smooth) functions approximating a semicontinuous function $f$.
Here, for approximation, pointwise convergence is fine.
For ...

6
votes

1
answer

456
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### Is the embedding dimension minus the dimension upper semicontinuous?

For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant $$\operatorname{def}(R) := \operatorname{dim}_K(\mathfrak{m}/\mathfrak{m}^2) - \...

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votes

1
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### Do semialgebraic sets depend outer semicontinuously on their defining polynomials?

Consider a compact (semialgebraic) ball $B\subset \mathbb{R}^n$ and a semialgebraic set $A=A(f_1,\cdots,f_s)\subset \mathbb{R}^n$ defined through some representation in terms of polynomials $f_1,\...

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### Upper semicontinuity in C(X)-algebras. Quotient norm question

upper semicontinuity in C(X)-algebras
In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$
By the definition of the quotient ...

7
votes

0
answers

482
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### Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...

12
votes

2
answers

701
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### Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:
Definition. Let $(X,d)$ be a ...

14
votes

6
answers

5k
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### More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.
Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...