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equicontinuous∗ asteroid† 2013-03-22 2:13:36 1 Definition Let X be a topological space, (Y, d) a metric space and C(X, Y ) the set of continuous functions X → Y . Let F be a subset of C(X, Y ). A function f ∈ F is continuous at a point x0 when given > 0 there is a neighbourhood U of x0 such that d(f (x), f (x0 )) < for every x ∈ U . When the same neighbourhood U can be chosen for all functions f ∈ F, the family F is said to be equicontinuous. More precisely: Definition - Let F be a subset of C(X, Y ). The set of functions F is said to be equicontinuous at x0 ∈ X if for every > 0 there is a neighbourhood U of x0 such that for every x ∈ U and every f ∈ F we have d(f (x), f (x0 )) < The set F is said to be equicontinuous if it is equicontinuous at every point x ∈ X. 2 Examples • A finite set of functions in C(X, Y ) is always equicontinuous. • When X is also a metric space, a family of functions in C(X, Y ) that share the same Lipschitz constant is equicontinuous. • The family of functions {fn }n∈N , where fn : R → R is given by fn (x) := arctan(nx) is not equicontinuous at 0. ∗ hEquicontinuousi created: h2013-03-2i by: hasteroidi version: h41376i Privacy setting: h1i hDefinitioni h54E35i h54C35i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 3 Properties • If a subset F ⊆ C(X, Y ) is totally bounded under the uniform metric, then F is equicontinuous. • Suppose X is compact. If a sequence of functions {fn } in C(X, Rk ) is equibounded and equicontinuous, then the sequence {fn } has a uniformly convergent subsequence. (Arzel’s theorem) • Let {fn } be a sequence of functions in C(X, Y ). If {fn } is equicontinuous and converges pointwise to a function f : X → Y , then f is continuous and {fn } converges to f in the compact-open topology. References [1] J. Munkres, Topology (2nd edition), Prentice Hall, 1999. 2