  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 the system
$\mathsf{WKL}_0$;
  (ii) Cousin's lemma for Baire 1 functions is at least as strong as
$\mathsf{ACA}_0$;
  (iii) Cousin's lemma for Baire 2 functions is at least as strong as
$\mathsf{ATR}_0$.
