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Title: Continuous Images of Compact Sets are Compact
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Series: Real Analysis
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Chapter: Continuous Functions
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YouTube-Title: Real Analysis 30 | Continuous Images of Compact Sets are Compact
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Bright video: https://youtu.be/6VWTG4wlRoA
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Dark video: https://youtu.be/WdDozI8S8mU
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra30_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps
00:00 Intro
00:14 A special property of continuous functions
01:09 Theorem about images of compact sets
03:25 Proof of the Theorem
06:33 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: A \rightarrow \mathbb{R}$ be continuous. Which statement is not correct in general?
A1: If $A \subseteq \mathbb{R}$ is compact, then $f[A] \subseteq \mathbb{R}$ is compact.
A2: If $A \subseteq \mathbb{R}$ is bounded, then $f[A] \subseteq \mathbb{R}$ is bounded.
A3: If $A \subseteq \mathbb{R}$ is finite, then $f[A] \subseteq \mathbb{R}$ is finite.
Q2: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Does the maximum of $f$ always exist?
A1: Yes!
A2: No!
Q3: Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous. Does the maximum of $f$ always exist?
A1: Yes!
A2: No!
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Last update: 2025-01
