Questions: Pointwise Convergence of Function Sequences

Question 1

Consider fₙ(x) = xⁿ on [0, 1]. The pointwise limit is f(x) = 0 for x ∈ [0,1) and f(1) = 1. What does this example illustrate?

• A. The convergence is uniform because all functions fₙ are continuous on [0,1]
• B. The pointwise limit f is discontinuous even though every fₙ is continuous — pointwise convergence does not preserve continuity
• C. The limit must be wrong because a sequence of continuous functions always converges to a continuous function
• D. Pointwise convergence cannot be defined on closed intervals

Answer: (B)

Each xⁿ is continuous on [0,1], but the pointwise limit has a jump discontinuity at x = 1 (it equals 0 just to the left and 1 at x = 1). This is the canonical example of pointwise convergence's pathological behavior: the limit of a sequence of continuous functions need not be continuous. The convergence is also not uniform — for any fixed n, points x close to 1 satisfy xⁿ close to 1, not close to 0, so no single N makes the approximation uniformly good near x = 1.

Question 2

In the definition of pointwise convergence of (fₙ) to f on S, the threshold N such that |fₙ(x) − f(x)| < ε for all n > N depends on:

• A. Only ε — once you fix a tolerance, the same N works for all x ∈ S
• B. Only x — different points converge at different speeds regardless of tolerance
• C. Both ε and x — the N needed to get within ε of the limit can vary across different points in S
• D. Neither ε nor x — N is determined by the sequence alone

Answer: (C)

This is the defining characteristic of pointwise (as opposed to uniform) convergence. For each fixed x, you ask how large n must be so that fₙ(x) is within ε of f(x) — and the answer can vary across different points. Some parts of the domain may converge quickly, others slowly. The logical quantifier order is: ∀x ∀ε ∃N(x,ε), where N is chosen after x is fixed. Uniform convergence flips this: ∀ε ∃N ∀x, so a single N works everywhere simultaneously.

Question 3

In the definition of pointwise convergence, the value of N is allowed to depend on the particular point x ∈ S being considered.

• T. True
• F. False

Answer: True

This is the defining feature that separates pointwise from uniform convergence. In pointwise convergence, for each x separately, you find an N that works for that specific x and ε — different points may need different N values. A slowly converging part of the domain can require a much larger N than a rapidly converging part. Uniform convergence is the stronger condition where a single N works simultaneously for all x, meaning the sequence converges at a rate that does not depend on location.

Question 4

If each function fₙ in a sequence is continuous on [a, b] and fₙ converges pointwise to f, then f must be continuous on [a, b].

• T. True
• F. False

Answer: False

False — fₙ(x) = xⁿ on [0,1] is the standard counterexample. Every xⁿ is continuous, but the pointwise limit is 0 on [0,1) and 1 at x = 1, which is discontinuous at x = 1. Continuity IS preserved under uniform convergence (a key theorem of real analysis), which is one reason uniform convergence is the analytically 'right' notion: it supports the interchange of limits and continuous operations that pointwise convergence cannot guarantee.

Question 5

Explain the key difference between pointwise and uniform convergence in terms of the logical structure of their definitions.

Model answer: In pointwise convergence, the required index N depends on both the tolerance ε and the specific point x: for each x, you independently find how large n must be to get within ε of the limit at that point. The logical structure is: ∀x ∀ε ∃N(x,ε) such that n > N implies |fₙ(x) − f(x)| < ε. In uniform convergence, N is found before x is chosen: ∀ε ∃N ∀x such that n > N implies |fₙ(x) − f(x)| < ε for all x simultaneously. Uniform convergence requires the sequence to converge at a rate that works uniformly across the entire domain.

The quantifier order — whether N is chosen before or after x — is the precise mathematical statement of this difference. It is not a technicality: it determines which analytic properties (continuity, integrability, differentiability) are preserved by the limit. Pointwise convergence is too weak to guarantee any of these interchanges; uniform convergence is exactly strong enough.