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Mathematics · lesson 2 of 3

Pointwise versus uniform convergence

The cutoff N may depend on both ε and the point x.

Advanced university8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Sequence limits and supremum norms.

Pointwise

The cutoff N may depend on both ε and the point x.

Uniform

One N must work for all points in the domain at once.

Supremum norm

Uniform convergence is equivalent to sup_x abs(f_n(x)−f(x)) tending to zero.

Continuity

A uniform limit of continuous functions is continuous; pointwise limits need not be.

Worked examples

Example 1

Find the pointwise limit of f_n(x)=x^n on [0,1].

Credit both the interior and the endpoint.

The limit is 0 for 0≤x<1 and 1 at x=1.

Example 2

Explain why x^n does not converge uniformly on [0,1].

Credit either the continuity theorem with hypotheses or a valid supremum-error argument.

Each x^n is continuous but the pointwise limit jumps at 1. A uniform limit of continuous functions would be continuous; equivalently the supremum error remains 1.

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 Find the pointwise limit of f_n(x)=x^n on [0,1].

Sample answer and marking guidance

The limit is 0 for 0≤x<1 and 1 at x=1.

Credit both the interior and the endpoint.

2 Explain why x^n does not converge uniformly on [0,1].

Sample answer and marking guidance

Each x^n is continuous but the pointwise limit jumps at 1. A uniform limit of continuous functions would be continuous; equivalently the supremum error remains 1.

Credit either the continuity theorem with hypotheses or a valid supremum-error argument.

3 For f_n(x)=x/n on [0,2], find sup abs(f_n) when n=8.

Show answer and explanation

0.25

The maximum occurs at x=2 and equals 2/8.

4 For uniform convergence, can the cutoff N depend on x?

Show answer and explanation

No

A single cutoff must work across the whole domain.

5 Show that x/n converges uniformly to zero on [0,2].

Sample answer and marking guidance

The supremum error is 2/n, which tends to zero. Given ε>0, choose N>2/ε.

Credit a domain-wide bound.

6 Does pointwise convergence of continuous functions preserve continuity automatically?

Show answer and explanation

No

The x^n example on [0,1] disproves the claim.

7 For a fixed 0<a<1, show x^n converges uniformly to zero on [0,a].

Sample answer and marking guidance

The supremum is a^n, which tends to zero because 0<a<1.

Credit the restricted domain and the geometric decay.

8 State why interchanging a limit and an integral on a finite interval is justified for a uniformly convergent sequence of continuous functions.

Sample answer and marking guidance

The integral of the difference is bounded in magnitude by interval length times the uniform error, which tends to zero.

Credit the explicit bound and the finite interval assumption.

Further reading and review references

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