Denjoy (1932) proved the theorem in the course of his topological classification of homeomorphisms of the circle.
Assume that it has positive derivative ƒ′(x) > 0 that is a continuous function with bounded variation on the interval [0,1).
If ƒ is a C2 map, then the hypothesis on the derivative holds; however, for any irrational rotation number Denjoy constructed an example showing that this condition cannot be relaxed to C1, continuous differentiability of ƒ. Vladimir Arnold showed that the conjugating map need not be smooth, even for an analytic diffeomorphism of the circle.
Later Michel Herman proved that nonetheless, the conjugating map of an analytic diffeomorphism is itself analytic for "most" rotation numbers, forming a set of full Lebesgue measure, namely, for those that are badly approximable by rational numbers.
His results are even more general and specify differentiability class of the conjugating map for Cr diffeomorphisms with any r ≥ 3.