List questions from semicontinuous-functions

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Lower semicontinuous submeasure is countably subadditive

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89

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Prove that a function is upper semi-continuous

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149

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How to remember which is lower/upper semicontinuity?

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237

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Lower semicontinuity and partial minimization

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44

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For a continuous function defined on [a,b] , is the set of points at which f(x)>d closed set?

Published on 23 Feb 2026 - 3:07

173

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$f$ is LSC at $x$ if and only if $\lim_{\delta \to 0}\inf\{f(y) | y \in B(x,\delta)\}=f(x)$

Published on 23 Feb 2026 - 3:09

322

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How to show lower semicontinuity: differentiability $\rightarrow$ continuity $\rightarrow$ lower semicontinuity?

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224

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Subharmonic on $U$ iff subharmonic on each $U_{\alpha}$, where $(U_{\alpha})$ is an open cover of $U$

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67

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One-Sided Notion of Topological Closure

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446

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prove a function is lower semicontinuous

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797

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Show that the function $h$ defined by $h(x)=g(f(x))$ is lower semicontinuous.

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95

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Convex sets and semicontinuous functions

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525

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A proof that the pointwise limits of lower semicontinuous (lsc) functions is lsc

Published on 23 Feb 2026 - 4:47

204

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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

Published on 23 Feb 2026 - 4:48

List Question

The reason for a certain requirement in upper-semicontinuity

Lower semicontinuous submeasure is countably subadditive

Prove that a function is upper semi-continuous

How to remember which is lower/upper semicontinuity?

Lower semicontinuity and partial minimization

For a continuous function defined on [a,b] , is the set of points at which f(x)>d closed set?

$f$ is LSC at $x$ if and only if $\lim_{\delta \to 0}\inf\{f(y) | y \in B(x,\delta)\}=f(x)$

How to show lower semicontinuity: differentiability $\rightarrow$ continuity $\rightarrow$ lower semicontinuity?

Subharmonic on $U$ iff subharmonic on each $U_{\alpha}$, where $(U_{\alpha})$ is an open cover of $U$

One-Sided Notion of Topological Closure

prove a function is lower semicontinuous

Show that the function $h$ defined by $h(x)=g(f(x))$ is lower semicontinuous.

Convex sets and semicontinuous functions

A proof that the pointwise limits of lower semicontinuous (lsc) functions is lsc

Show a function mapping a metric space to $\mathbb{R}$ is continuous if and only if it is both upper and lower semi-continuous

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