It is detailed in the article itself. Cantor wanted to say what classes of sets of discontinuities are allowed such that we can still represent a function with

It is detailed in the article itself. Cantor wanted to say what classes of sets of discontinuities are allowed such that we can still represent a function with a trigonometric series. Cantor's approach immediately led to what are now called Cantor-Bendixson derivatives, and indirectly led to Cantor developing the rudiments of modern (naive) set theory.