List of basic formulas and concepts related to uniform convergence:
- Definition of uniform convergence:
A sequence of functions ( f_n(x) ) converges uniformly to a function ( f(x) ) on a set ( D ) if for any ( \epsilon > 0 ) there exists ( N ) such that for all ( n \geq N ) and for all ( x \in D ) the following holds:
[
|f_n(x) – f(x)| < \epsilon
] - Criterion of uniform convergence (according to Cauchy):
A sequence of functions ( f_n(x) ) converges uniformly on a set ( D ) if for any ( \epsilon > 0 ) there exists ( N ) such that for all ( m, n \geq N ) and for all ( x \in D ) the following holds:
[
|f_n(x) – f_m(x)| < \epsilon
] - Uniform convergence theorem:
If a sequence ( f_n(x) ) converges uniformly to ( f(x) ) on a set ( D ), and every function ( f_n ) is continuous on ( D ), then the limit function ( f(x) ) is also continuous on ( D ). - Example:
If ( f_n(x) = \frac{x}{n} ) on the interval ( [0, 1] ), then:
[
\lim_{n \to \infty} f_n(x) = 0
]
and this convergence is uniform, since for any ( \epsilon > 0 ) one can choose ( N = \lceil \frac{1}{\epsilon} \rceil ) so that for all ( n \geq N ) and all ( x \in [0, 1] ) ( |f_n(x)| < \epsilon ) holds.