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

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