• Title: Continuous Images of Compact Sets are Compact

  • Series: Real Analysis

  • Chapter: Continuous Functions

  • YouTube-Title: Real Analysis 30 | Continuous Images of Compact Sets are Compact

  • Bright video: https://youtu.be/6VWTG4wlRoA

  • Dark video: https://youtu.be/WdDozI8S8mU

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ra30_sub_eng.srt missing

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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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