# Questions tagged [semicontinuous-functions]

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

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### Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
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### Is this function upper-semi continuous?

Let $(A,\mathcal A)$ be a measure space, let $\mathcal P(A)=\{ q : (A, \mathcal A, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(A,\mathcal A)$. We equip $\mathcal P(A)$ with ...
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### Counter example for: If $f\colon \mathbb{R}^n\to \mathbb{R}$ lower semi continuous, then $\forall x_n\to x, \lim_{n\to\infty}f(x_n)\geq f(x)$

We start with a definition. We say that a function $f\colon \mathbb{R}^n\to \mathbb{R}$ is lower semi continuous (l.s.c.) if $$\forall x\in\mathbb{R}^n,\; \liminf_{y\to x}f(y)\geq f(x).$$ As ...
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### $\lim_{k \rightarrow \infty} \frac{f(x)}{k}$ where $f: X \rightarrow \mathbb{R} \cup \{\infty\}$

I have this line of assumptions for some exercise to be proven "Let $X$ be a metric space and let $f: X \rightarrow \mathbb{R} \cup \{+\infty\}$ be a lower semi-continuous function which is ...
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### Lower semicontinuity of the rank for a map of vector bundles

In Nitsure's Cohomology of the moduli of parabolic vector bundles, we have the following Remark on page $62$: ([Remark 1.2) Let $E$ be a vector bundle on a scheme $S$ and let $F_1$ and $F_2$ be ...
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### Measure of closed set is vaguely u.s.c. on M_1^+(K)

I'm working my way through Compact convex sets and boundary integrals by Alfsen and during the proof of Proposition I.2.8 they use that set $W=\{\mu\in M_1^+(K)|\mu(\overline{V})<\alpha\}$ is ...
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### $f$ is lower semi-continuous at $x_0$ if and only if $\liminf f(x_k) \geq f(x_0)$ for all $x_k \rightarrow x_0$

Prove the following claim:$f$ is lower semi-continuous at $x_0$ if and only if $\liminf_{k\rightarrow \infty} f(x_k) \geq f(x_0)$ for all $x_k \rightarrow x_0$ as $k\rightarrow \infty$. First, assume ...
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### Prove that $\Phi$ is a lower semicontinuous functional.

QUESTION: Let $(X, \|.\|)_X$ be a Banach space and $Y\subset X$ a subspace, which is itself a Banach space endowed with a norm $\|.\|_Y$ such that $\|y\|_X\leq \|x\|_Y$ for every $y\in Y$. Assume that ...
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### Is the limit of an increasing sequence of continuous functions is a lower semicontinuous function [closed]

1.$f_n(x)$ is continuous，bounded，positive and $f_n(x)\le f_{n+1}(x)$ 2.$f(x)=\lim f_n(x)$ Question：Is $f(x)$ a lower semicontinuous function？
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### Is there any function $f:\mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}$ such that $f$ is lower semi continuous only at $\mathbb{Q}$?

Is there any function $f:\mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}$ such that $f$ is lower semi continuous only at $\mathbb{Q}$ ? A function $f:\mathbb{R} \to \mathbb{R}\cup \{\pm \infty\}$ is said ...
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### Finite valued, semi-continuous functions on a measure space: is it locally constant almost everywhere?

Let $f:X\to [\![a,b ]\!]$ be a discrete-valued, lower-semicontinuous function on an arbitrary Borel measure space $X$. Let $A=\{x \in X: f \text{ is not locally constant at } x\}$ Question: does $A$ ...
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### Definition of Semicontinuity: Confusion in Rudin's RCA

Rudin's RCA defines upper and lower semicontinuity as follows: Let $f$ be a real or extended-real function on a topological space. If $$\{x: f(x) > \alpha\}$$ is open for every real $\alpha$, $f$ ...
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### Do I have the wrong interpretation of the limit inferior in this definition?

I am trying to understand the definition of lower semi-continuity. A function $f$ is lower semi-continuous at some point $x_0$ if the following holds $$\lim_{x \to x_0} \inf f(x) \geq f(x_0).$$ This ...
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### How to prove the criteria for lower semicontinuity of a function at a point?

I have the following statement I need to prove. Let $f:D \to \mathbb{R}$. Then the function $f$ is lower semicontinuous at some point $x_0 \in D$ if $$\lim_{x \to x_0 } \inf f(x) \geq f(x_0)$$. I have ...
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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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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 ...
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, ...