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Title: Improper Riemann Integrals of Unbounded Functions
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 63 | Improper Riemann Integrals of Unbounded Functions
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Exercise Download PDF sheets
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Subtitle on GitHub: ra63_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: [0,1] \rightarrow \mathbb{R}$ be Riemann-integrable. Which conclusion is correct?
A1: $f$ is continuous.
A2: $f$ is monotonic.
A3: $f$ is bounded.
Q2: $f: [0,1] \rightarrow [0,1]$ be given by $$ f(x) = \begin{cases} x^2, & x > 0 \ 1, & x = 0 \end{cases}$$ Is this function Riemann-integrable?
A1: Yes!
A2: No!
A3: Only as an improper Riemann integral.
Q3: $f: [0,1] \rightarrow [0,1]$ be given by $$ f(x) = \begin{cases} \frac{1}{x}, & x > 0 \ 1, & x = 0 \end{cases}$$ Does the integral $$\int_0^1 f(x) , dx$$ exist?
A1: No, the improper Riemann integral does not converge.
A2: Yes, it’s a well-defined Riemann integral.
A3: Yes, but only as an improper Riemann integral.
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Last update: 2025-01
