Mathematics · Advanced university

Pointwise versus uniform convergence

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

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Prerequisites

Sequence limits and supremum norms.

Study notes

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

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.

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

For f_n(x)=x/n on [0,2], find sup abs(f_n) when n=8.
For uniform convergence, can the cutoff N depend on x?
1. Only when x is rational
2. Yes, without restriction
3. Only at positive x
4. No
Show that x/n converges uniformly to zero on [0,2].
Does pointwise convergence of continuous functions preserve continuity automatically?
1. No
2. Only because each term is bounded
3. Yes in every domain
4. The limit must be a polynomial

Answers and marking guidance

Answers and marking notes

0.25
The maximum occurs at x=2 and equals 2/8.
No
A single cutoff must work across the whole domain.
The supremum error is 2/n, which tends to zero. Given ε>0, choose N>2/ε.
Credit a domain-wide bound.
No
The x^n example on [0,1] disproves the claim.