Coding of real-valued continuous functions under WKL $\mathsf {WKL}$

Abstract

In the context of constructive reverse mathematics, we show that weak Kőnig's lemma ($\mathrm{WKL}$) implies that every pointwise continuous function $f:[0,1]\to R$ is induced by a code in the sense of reverse mathematics. This, combined with the fact that $\mathrm{WKL}$ implies the Fan theorem, shows that $\mathrm{WKL}$ implies the uniform continuity theorem: every pointwise continuous function $f:[0,1]\to R$ has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier-free axiom of choice.