The Notorious Jumping Function of Continuum County

Submitted by Deb from Rochester, NY, 2/26/2000. Original answer and this article by Allen Stenger.

Give an example of monotonic function on [0,1] which is discontinuous at all rationals.

Hint 1

Sneak up on it a few discontinuities at a time.

Hint 2

We can enumerate the rationals in the interval [0,1] in groups as

0/1, 1/1
1/2
1/3, 2/3
1/4, 3/4

and so on. For each group, invent a monotonic function that is discontinuous at each point of the group. Then what?

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