Cousin’s lemma in second-order arithmetic

Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $\mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to $\mathsf{WKL}_0$; (ii) Cousin's lemma for Baire class 1 functions is equivalent to $\mathsf{ACA}_0$; (iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to $\mathsf{ATR}_0$ (modulo some induction).