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Math and science::Analysis::Tao::09. Continuous functions on R

Uniform continuity (sequence dual)

Uniform continuity: 2/3

ε-δ vs. sequence formulation

Set adherence and function convergence have both an "epsilon-delta" formulation and a sequence formulation. We covered propositions that mapped between these equivalences.

Uniform convergence also has this duality. Below, uniform convergence is formulated in terms of sequences. The sequences equivalence formulations below look very similar to the formulation of equivalent Cauchy sequences, but they are not necessarily Cauchy, and that is their distinction.

Equivalent sequences

Let \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) be sequences of reals. We say that \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) are equivalent sequences iff for any real \( \varepsilon > 0 \) there exists an integer \( N \ge m \) such that \( |a_k - b_k| < \varepsilon \) for all \( k \ge N \).

This formulation can also be split up by using intermediate definitions for ε-close sequences and eventual ε-close sequences.

The above definition can be succinctly formulated using sequence limits also.

Equivalent sequences (via limits)

Let \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) be sequences of reals. We say that \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) are equivalent sequences iff [...].

Now on to uniform convergence phrased using equivalent sequences.

Uniform convergence (via equivalent sequences)

Let [...] be a set and let [...] be a function. Then the following two statements are logically equivalent:

  • f is uniformly continuous.
  • [...]

Useful properties

Three useful properties concerning uniformly continuous functions are:

  • they map [some type of sequence] to [some type of sequence].
  • they map [some type of set] to [some type of set].
  • If a fuction \( f \) has a [a something domain] and is [...] then it is also uniformly continuous.

The last of these is proved by Tao; the others are exercises.

Mapping of sequences

Function convergence and sequence convergence

The duality of function convergence and sequence convergence means that if \( f \) is a continuous function, then \( f \) maps [...] to [...].

Uniform function convergence and pairs of equivalent sequences

The duality of uniform function convergence and pairs of equivalent sequences means that if \( f \) is a uniformly convergent function, then \( f \) maps [...] to pairs of [...].

Connecting the two

To see the connection between the two ideas above, observer that a sequence \( (a_n)_{n=0}^{\infty} \) converges to \( x \) if and only if the sequence \( (a_n)_{n=0}^{\infty} \) is equivalent to [...].