Question

Q2. Let f:(a,b) → R be a monotonic function. Then the set of points of (a, b) where f is discontinuous
is at most countable.
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Answer 1

Answer:

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

please mark this brainliest

Answer 2

The result I am trying to prove is the following:

Let F:R→R be increasing. Then the set of points, at which it is discontinuous, is countable.

I have been reading this form Folland's Real Analysis and he proves this by considering the sum ∑|x|<N[F(x+)−F(x−)] which has to be finite by some sort of telescoping sum. Now, I am not sure how he defines this sum. I am assuming he does this by using nets (let me know if there is another way to look at this), but he doesn't introduce the concept of nets until the next chapter. So I was looking for some other ways to prove this result and I found the following result (from Stein-Shakarchi) which is quite similar but assumes F to be bounde