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

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