This is a question in our book, and the answer in the teacher's book is "No", and a friend of mine says that their teacher also says that a rational function always has a definite number of docontinuity points.
But I can think of a few functions that have an infinite number of discontinuity points;
1) which is discontinuous at every value that satisfies where is an integer. The same is also true for this function but with cosine and tan and their respective infinite sets of discontinuity points
2) which is discontinuous over the whole interval and has an infinte number of discontinuity points that belong to this interval
Now I think I'm wrong here because I'm not sure if these count as "rational functions" or not.