# Questions tagged [semicontinuity]

The semicontinuity tag has no usage guidance.

24
questions

**-1**

votes

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53 views

### Why is $f$ lower semicontinuous and convex? [closed]

Let $\mathcal{H}$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and $L \in \mathcal{B(H)}$ be self-adjoint and monotone. Let $ u\in \mathcal{H}, \alpha \in \mathbb{R}$. set
$$ ...

**1**

vote

**1**answer

78 views

### 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

52 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**

votes

**0**answers

78 views

### 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**

votes

**2**answers

193 views

### 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. ...

**3**

votes

**1**answer

293 views

### 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

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151 views

### 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

81 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(...

**1**

vote

**1**answer

72 views

### 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**

votes

**1**answer

152 views

### 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

148 views

### 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

78 views

### 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

112 views

### 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

231 views

### 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

645 views

### 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**

vote

**0**answers

27 views

### 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 \...

**6**

votes

**2**answers

2k views

### 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

107 views

### 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

586 views

### 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 ...

**5**

votes

**1**answer

325 views

### 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) - \...

**8**

votes

**1**answer

222 views

### 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,\...

**1**

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**0**answers

62 views

### 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

432 views

### 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 ...

**10**

votes

**2**answers

553 views

### 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 ...