# Questions tagged [semicontinuous-functions]

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

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### 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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### 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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### 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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### 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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### 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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### 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 ...
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: ...
### 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 ...