Mathematics · Advanced university
Compactness and continuous functions · revision notes
In R^n, compactness is equivalent to being closed and bounded.
Key ideas and conditions
Euclidean criterion
In R^n, compactness is equivalent to being closed and bounded.
Extreme values
A continuous real-valued function on a nonempty compact set attains a maximum and minimum.
Uniform continuity
Continuity on a compact domain implies uniform continuity.
Domain matters
Closed-and-bounded is not a general compactness criterion in every metric or normed space.
Explain without looking
- Does a continuous image of a compact set remain compact?
- Distinguish a supremum from a maximum.
Further reading and review references
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