{"id":424,"date":"2016-10-20T21:08:48","date_gmt":"2016-10-21T01:08:48","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=424"},"modified":"2017-08-22T17:01:17","modified_gmt":"2017-08-22T21:01:17","slug":"sets-of-discontinuity","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2016\/10\/20\/sets-of-discontinuity\/","title":{"rendered":"Sets of Discontinuity"},"content":{"rendered":"

In Chapter 4, we spent much of our time trying to understand and prove continuity of functions. However, as we saw in Section 4.1, discontinuous functions can be even more interesting. Recall the Dirchlet Function, which we saw was discontinuous on all of \"\mathbb{R}\", or well as the modified Dirchlet, which was discontinuous on \"\mathbb{R}\ , and don\u2019t forget the Thomae Function, which was discontinuous on all of \"\mathbb{Q}\". What is going on with these discontinuities, and what does this tell us about the discontinuities of an arbitrary function?<\/p>\n

To start, let\u2019s define, for a function \"f:, the set \"D_f as the set of all points where \"f fails to be continuous. It turns out that this set has a number of really interesting properties for a generic function on the real numbers. In general, we can divide discontinuities of a function at a point \"c\" into three categories:<\/p>\n