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Title: Pointwise Convergence
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Series: Real Analysis
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Chapter: Continuous Functions
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YouTube-Title: Real Analysis 24 | Pointwise Convergence
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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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: ra24_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
00:33 Pointwise convergence
02:03 1st Example
03:20 2nd Example
06:40 3rd Example
07:43 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: I \rightarrow \mathbb{R}$. What is the correct definition for the pointwise convergence to a function $f: I \rightarrow \mathbb{R}$?
A1: $ \displaystyle \forall x \in I ~~ \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $
A2: $ \displaystyle \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall x \in I ~~ \forall n \geq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $
A3: $ \displaystyle \forall x \in I ~~ \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \geq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $
A4: $ \displaystyle \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall x \in I ~~ \forall n \leq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $
Q2: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$ given by $f_n(x) = \frac{x}{n^2} - 2 - \frac{1}{n} $. What is the pointwise limit $f$?
A1: $ \displaystyle f(x) = \frac{x}{n^2} - 2 - \frac{1}{n} $
A2: $ \displaystyle f(x) = 0 $
A3: $ \displaystyle f(x) = x - 2 $
A4: $ \displaystyle f(x) = \frac{x}{2} - 2 - \frac{1}{2} $
A5: $ \displaystyle f(x) = -2 $
Q3: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$. Which example does not have a pointwise limit?
A1: $ \displaystyle f_n(x) = \frac{x}{n^2} - 2 - \frac{1}{n} $
A2: $ \displaystyle f_n(x) = x - 2$
A3: $ \displaystyle f_n(x) = \begin{cases} n^2 x - 2 &, ~~ x = 0\ x - \frac{1}{n} &, ~~x > 0\end{cases} $
A4: $ \displaystyle f_n(x) = \frac{n^2 x}{2} - 2 $
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Last update: 2025-01
