Some topological properties of the function ๐(๐ฆ)
S. Sawyer ยท Proceedings of the American Mathematical Society ยท 1967
for the total variation of f(x). In particular, if f(x) is of bounded variation it has an a.e. finite n(y), but in general we could have n(y) = oo identically in the range of f(x) (see e.g. ?3). The condition T1 that n(y) < oo a.e. has many real variable characterizations ([2], [4, pp. 278-287]); for example, it is equivalent to certain differentiability conditions on f(x). Nina Bary [2] has also shown that any continuous function on the unit interval can be written as the sum of three continuous functions satisfying T1. The paper [3] is in a slightly different vein, in which an a.e. equivalent definition of n(y) is given with applications in Fourier series. For example (assuming f(O) =f(1)), the condition