This is a question in our book, and the answer in the teacher's book is "No", and a friend of mine s

babyagelesszj 2022-07-15 Answered
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 n number of docontinuity points.
But I can think of a few functions that have an infinite number of discontinuity points;
1) f ( x ) = 1 sin x which is discontinuous at every x value that satisfies x = n π where n 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) f ( x ) = 1 r 2 x 2 which is discontinuous over the whole interval [ r , r ] 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.
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Answers (1)

Jayvion Mclaughlin
Answered 2022-07-16 Author has 14 answers
Actually, since a rational function is a quotient of two polynomial functions, since polynomial functions are continuous and since the quotient of two continuous functions is continuous, every rational function has zero points of discontinuity.
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