Questions tagged [semicontinuous-functions]

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

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

Some authors us the name semicontinuity also for multifunctions, many authors use the name hemicontinuous. This tag is specifically for questions about functions, for multifunctions use .

173 questions
26
votes
4 answers

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 $\forall \alpha\in R$, the set $\{x\in X : f(x) > \alpha \}$ is open in…
mononono
  • 1,848
  • 15
  • 22
15
votes
1 answer

Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist an increasing sequence of $k$-times continuously…
12
votes
1 answer

Upper semi continuous, lower semi continuous

which of the followings are true? $X$ be a topological space, $f_n:X\rightarrow \mathbb{R}$ is sequence of lower semi continuous functions then the $\sup\{f_n\}=f$ is also lower semi continuous. every continuous real valued function on $X$ is…
Marso
  • 31,275
  • 18
  • 107
  • 243
12
votes
1 answer

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 $X$. Define $$g_k=\sup_{n\ge k}f_n.$$ I could see that…
12
votes
2 answers

Any lower semicontinuous function $f: X \to \mathbb{R}$ on a compact set $K \subseteq X$ attains a min on $K$.

I've been thinking about this problem for a long time right now, and feel stuck. Given that $X$ is a topological space, and that for $f$ to be lower semicontinuous, for any $x \in X$ and $\epsilon > 0$, there is a neighborhood of $x$ such that…
MathNewbie
  • 1,453
  • 13
  • 25
11
votes
5 answers

Lower semicontinuous function as the limit of an increasing sequence of continuous functions

Let $ f:\mathbb{R}^m \rightarrow (-\infty,\infty] $ be lower semicontinuous and bounded from below. Set $f_k(x) = \inf\{f(y)+k d( x,y ): y\in \mathbb{R}^m\} $ , where $d(x,y)$ is a metric. It is easy to see that each $f_k$ is continuous and $f_1 …
kes
  • 111
  • 1
  • 3
10
votes
1 answer

Show that every upper semi-continuous real function is measurable

Possible Duplicate: Subset of the preimage of a semicontinuous real function is Borel A real function $f$ on the line is upper semi-continuous at $x$, if for each $\epsilon > 0$, there exists $\delta > 0$ such that $|x-y|<\delta$ implies that…
9
votes
1 answer

Lower Semicontinuity Concepts

Let $X$ be a real Banach space, let $f:X\rightarrow \overline{\mathbb{R}}$ be a functional. We have known that: If $f$ is weakly lower semicontinuous then $f$ is weakly sequentially lower semicontinuous; If $f$ is weakly sequentially lower…
8
votes
1 answer

Subset of the preimage of a semicontinuous real function is Borel

I'm in a jam with this problem: Let $ f: \mathbb{R} \to [-\infty,\infty] $ be lower semicontinuous, and let $ A = \{ x:f(x)\ge a \} $. Is $A$ necessarily a Borel set in $ \mathbb{R} $? I've actually managed to prove that if $A$ has no excluded…
8
votes
1 answer

Equivalence of definitions for upper semicontinuity

I am trying to show that a function is upper semicontinuous if and only if the preimage of any open ray $(-\infty, a)$ is open. The definition given for upper semicontinuity is that $\lim\limits_{k \to \infty} x_k = x \implies \limsup\limits_{k\to…
Aden Dong
  • 1,337
  • 7
  • 20
8
votes
2 answers

Show the iff statement of lower semicontinuous

Given $X\subseteq \Bbb R^m, f:X\to\Bbb R$ and $x\in X$, we say $f$ is lower semicontinous (l.s.c for short) at x if$\forall \varepsilon>0\ \exists\ \delta >0\ \forall \in B(\delta,x), \ f(x)\le f(y)+\varepsilon$. I wish to show: If $X$ is closed,…
Scorpio19891119
  • 2,235
  • 3
  • 25
  • 40
8
votes
1 answer

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 $f_\alpha$, $\alpha \in I$ be a collection of lower…
user438666
  • 1,930
  • 1
  • 8
  • 19
7
votes
1 answer

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 $g(x)=\inf _{1 \leq k<\infty} f_{k}(x)$. Show that $g$ is upper…
6
votes
1 answer

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 with restrictions like Banach-spaces or…
6
votes
2 answers

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 $f\colon \mathbb{R} \to \mathbb{R}$ is a smooth function, bounded above and below…
1
2 3
11 12