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Mathematics · Advanced university

Compactness and continuous functions

In R^n, compactness is equivalent to being closed and bounded.

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Prerequisites

Real limits, closed sets and boundedness.

Study notes

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.

Worked examples

Which subset of R is compact?

The interval is closed and bounded in R.

[0,1]

Give a continuous bounded function on (0,1) that does not attain its supremum.

Credit the open domain and absence of a maximizing point.

f(x)=x has supremum 1, but x=1 is not in the domain.

Check your understanding

  1. What hypothesis is missing from the claim that every continuous function has a maximum?

    1. An appropriate nonempty compact domain
    2. Only a name for the function
    3. A colorful graph
    4. A positive derivative everywhere

  2. Explain why f(x)=1/x is not uniformly continuous on (0,1).

  3. Find the maximum of x(1−x) on [0,1].

  4. What does sequential compactness say in a metric space?

Answers and marking guidance

Answers and marking notes
  1. An appropriate nonempty compact domain

    Continuity alone is insufficient on arbitrary domains.

  2. Take x_n=1/(n+1) and y_n=1/n for n≥2. Their distance tends to zero but abs(f(x_n)−f(y_n))=1.

    Credit two valid domain sequences with input distance tending to zero and output separation not tending to zero.

  3. 0.25

    Complete the square: x(1−x)=1/4−(x−1/2)².

  4. Every sequence in the set has a subsequence converging to a point in the set.

    Credit the subsequence and the requirement that its limit remain in the set.

Further reading and review references

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