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Title: Combination of Continuous Functions
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Series: Real Analysis
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Chapter: Continuous Functions
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YouTube-Title: Real Analysis 29 | Combination of Continuous Functions
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Bright video: https://youtu.be/W-E4LqZyEHA
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Dark video: https://youtu.be/aHwX1RVKLiA
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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: ra29_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 Recalling continuity
01:25 Combining 2 continuous functions
03:07 Composition of functions
05:55 Proof for composition of functions
07:27 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous at $x_0$. Which statement is not correct in general?
A1: The function $f+g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.
A2: The function $f-g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.
A3: The function $f \cdot g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.
A4: The function $f \circ g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.
Q2: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. At which points is the function $f/g$ continuous in general?
A1: At all points $x_0 \in \mathbb{R}$.
A2: At no point $x_0 \in \mathbb{R}$.
A3: At the points $x_0 \in \mathbb{R}$ where $g(x_0) \neq 0$.
A4: At $x_0 = 0$.
Q3: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Which statement is correct?
A1: $f \circ g$ is not well-defined in general.
A2: $f \circ g$ is a continuous function.
A3: There are some points where $f \circ g$ is not continuous.
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Last update: 2025-01
