# Questions tagged [semicontinuity]

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### Are hyperplanes Upper and Lower semicontinous correspondence? [closed]

Let $\mathcal{\Phi}:\mathbb{R}^2 \to P(\mathbb{R}^2)$, where $P(\cdot)$ is a power set, defined by $\mathcal{\Phi}(d) = \{x\in \mathbb{R}^2: d^Tx=0\}$ where $d^T$ is the transpose of vector $d$. So, ...
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### Sum of upper semi continuous and lower semi continuous functions

Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
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### Problem in understanding maximum principle for subharmonic functions

I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is. ...
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### 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. ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...