# 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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### Riemann integrability of upper semicontinuous functions [closed]

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