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A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators.
In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions.It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively ...
If the domain of the functions is a topological space and the codomain is a uniform space, local uniform convergence (i.e. uniform convergence on a neighborhood of each point) and compact (uniform) convergence (i.e. uniform convergence on all compact subsets) may be defined. "Compact convergence" is always short for "compact uniform convergence ...
Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability.
(Monotone convergence theorem) If is bounded and monotonic for all greater than some , then it is convergent. A sequence is convergent if and only if every subsequence is convergent. If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.
Local uniform convergence (i.e. uniform convergence on a neighborhood of each point) Compact (uniform) convergence (i.e. uniform convergence on all compact subsets) further instances of this pattern below. Implications: - "Global" modes of convergence imply the corresponding "local" and "compact" modes of convergence. E.g.:
Note that () is continuous on the real closed interval [,] for <, by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that the restriction of G ( z ) {\displaystyle G(z)} to [ 0 , 1 ] {\displaystyle [0,1]} is continuous.