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Mathematics · lesson 3 of 3

Compactness and continuous functions

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

Advanced university8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Real limits, closed sets and boundedness.

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

Example 1

Which subset of R is compact?

The interval is closed and bounded in R.

[0,1]

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Which subset of R is compact?

Show answer and explanation

[0,1]

The interval is closed and bounded in R.

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

Sample answer and marking guidance

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

Credit the open domain and absence of a maximizing point.

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

Show answer and explanation

An appropriate nonempty compact domain

Continuity alone is insufficient on arbitrary domains.

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

Sample answer and marking guidance

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.

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

Show answer and explanation

0.25

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

6 What does sequential compactness say in a metric space?

Sample answer and marking guidance

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.

7 Does a continuous image of a compact set remain compact?

Show answer and explanation

Yes

Continuous images preserve compactness.

8 Distinguish a supremum from a maximum.

Sample answer and marking guidance

A supremum is the least upper bound. A maximum is an attained value that is at least every other value; the supremum need not be attained.

Credit the attainment distinction and a coherent definition.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.