Sequences of Functions — Mathematics Knowledge System

Overview

When infinitely many functions f_n converge to a limit function f, the mode of convergence matters profoundly. Pointwise convergence allows the limit to lose continuity or fail to commute with integration. Uniform convergence, a stronger condition, preserves these properties and is the key hypothesis in theorems that justify swapping limits with integrals and derivatives.

Intuition

Pointwise convergence means: for each fixed x, the sequence f_n(x) eventually gets close to f(x). The speed can vary wildly with x. Uniform convergence means: one N works for all x simultaneously -- the entire graph of f_n is within epsilon of f's graph for large n. Uniformity prevents the convergence from 'deteriorating' near certain points and lets us trust that integrals and continuity survive the limit.

Formal Definition

Definition

Let f_n and f be functions on a set E. The sequence (f_n) converges pointwise to f if for each x in E and each epsilon > 0 there exists N (depending on x and epsilon) with |f_n(x) - f(x)| < epsilon for all n >= N. It converges uniformly if N can be chosen independently of x.

f_n \to f \text{ pointwise} \iff \forall x,\, \forall \varepsilon>0,\, \exists N : n \geq N \Rightarrow |f_n(x)-f(x)| < \varepsilon

Pointwise convergence

f_n \to f \text{ uniformly} \iff \forall \varepsilon>0,\, \exists N : n \geq N \Rightarrow \sup_x|f_n(x)-f(x)| < \varepsilon

Uniform convergence

\sum_{n=1}^\infty M_n < \infty \text{ and } |f_n(x)| \leq M_n \Rightarrow \sum f_n \text{ converges uniformly}

Weierstrass M-test

Notation

Notation	Meaning
$f_n \xrightarrow{\text{pw}} f$	Pointwise convergence of sequence of functions
$f_n \rightrightarrows f$	Uniform convergence of sequence of functions
$\\|f\\|_{\infty} = \sup_x \|f(x)\|$	Supremum norm (uniform norm)

Theorems

Theorem 1: Uniform Limit of Continuous Functions is Continuous

If each f_n is continuous on E and f_n converges uniformly to f on E, then f is continuous on E. (Pointwise convergence alone does not guarantee this.)

Theorem 2: Uniform Convergence and Integration

If f_n converges uniformly to f on [a,b] and each f_n is Riemann integrable, then f is integrable and the integral of f equals the limit of the integrals of f_n: lim integral(f_n) = integral(f).

Theorem 3: Weierstrass M-Test

Let sum M_n be a convergent series of positive constants. If |f_n(x)| <= M_n for all x in E and all n, then sum f_n converges uniformly (and absolutely) on E.

Theorem 4: Power Series Converge Uniformly on Compact Subsets

A power series sum a_n*(x-c)^n with radius of convergence R converges uniformly on every compact interval [c-r, c+r] with r < R.

Worked Examples

1
Pointwise limit: for x in [0,1), x^n -> 0; for x = 1, x^n = 1 -> 1. So f(x) = 0 for x in [0,1) and f(1) = 1.
$f(x) = \lim_{n\to\infty} x^n = \begin{cases} 0 & x \in [0,1) \\ 1 & x=1 \end{cases}$
2
The limit function f is discontinuous at x = 1 even though each f_n is continuous. By Theorem 1, the convergence cannot be uniform.
3
Directly: sup_{x in [0,1]} |x^n - f(x)| = sup_{x in [0,1)} x^n = 1 (approached but never reached), so the sup norm does not go to 0.
$\sup_{x \in [0,1]} |f_n(x) - f(x)| = 1 \not\to 0$

✓ Answer

x^n converges pointwise on [0,1] but not uniformly, because the limit function is discontinuous.

Practice Problems

Mediumproof writing

Prove that if f_n -> f uniformly on [a,b] and each f_n is integrable, then lim integral_a^b f_n(x) dx = integral_a^b f(x) dx.

Mediumapplication

Does f_n(x) = sin(nx)/sqrt(n) converge uniformly on R?

Mediumapplication

The power series for e^x converges uniformly on [-2,2]. Justify this using the M-test.

Common Mistakes

Common Mistake

Pointwise convergence is enough to swap limits and integrals.

Pointwise convergence alone does not justify lim integral f_n = integral (lim f_n). You need uniform convergence (or dominated convergence in measure theory).

Common Mistake

If f_n -> f uniformly and each f_n is differentiable, then f is differentiable and f_n' -> f'.

Uniform convergence of f_n does not imply anything about derivatives. You need uniform convergence of the derivatives f_n' to conclude f' = lim f_n'.

Quiz

Historical Background

In the early 19th century, mathematicians routinely interchanged limits and integrals without justification, leading to errors. Cauchy believed continuity was preserved under pointwise limits, but Niels Abel found counterexamples around 1826. Karl Weierstrass clarified the distinction between pointwise and uniform convergence in the 1840s, providing the rigorous framework and the M-test that bears his name. This work was essential to placing analysis on a firm foundation.

1821
Cauchy incorrectly claims limits of continuous functions are continuous
Augustin-Louis Cauchy
1826
Abel notes counterexamples to Cauchy's claim via Fourier series
Niels Henrik Abel
1848
Weierstrass introduces uniform convergence and proves the M-test
Karl Weierstrass
1872
Weierstrass constructs a continuous but nowhere-differentiable function via uniform limits
Karl Weierstrass

Summary

Pointwise convergence: for each x, f_n(x) -> f(x); the speed can vary with x.
Uniform convergence: the supremum norm ||f_n - f|| -> 0; N works for all x simultaneously.
Uniform convergence preserves continuity and allows interchange of limits and integrals.
The Weierstrass M-test: if |f_n(x)| <= M_n and sum M_n < infty, then sum f_n converges uniformly.

References

BookRudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.
BookAbbott, S. (2015). Understanding Analysis (2nd ed.). Springer.
WebsiteWeisstein, E. W. Uniform Convergence. MathWorld.