Cousin’s lemma is a principle of compactness of the unit interval. It is a crucial tool in the development of the gauge integral (also known as the Kurzweil-Henstock-Denjoy-Perron integral), which is a common generalisation of both the Riemann and the Lebesgue integral. As expected for a principle of compactness, the restriction of Cousin’s lemma to continuous functions is equivalent to weak König’s lemma. The situation becomes more interesting when we consider Baire class 1 functions (pointwise limits of continuous functions), Baire class 2 functions, and general Borel functions. Joint work with Rod Downey and Jordan Barrett.