How Can Monotone Functions Have Only Countably Many Discontinuities?

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That means that the intervals (a_i, b_i) and (a_j, b_j) do not overlap. So it is possible to choose a rational number yi from (a_i, b_i) and (a_j, b_j) that is different from yj. But, since a_i< b_i, that means that there is a rational number in (a_i, b_i).
  • #1
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i have a function f:R->R where f is monotone increasing, i need to show that the set of discontinuous points of f is at most countable.
so i need to find an injective or 1-1 mapping from this set to the naturals, or to the rationals.
i thought perhaps defining the next function g:A->Q, where A is the set of discontinuous points of f, by:
let x0 be in A, so lim(x>x0)f(x)>lim(x<x0)f(x)
g(x0)=x0 if x0 is in Q
but how do i define for points which arent in Q?

thanks in advance.
 
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  • #2
First, the correct phrase is "points of discontinuity of f", not "discontinuous points". Points are not "continuous" or "discontinuous", only functions are!

Have you already shown that a discontinuity of an increasing function must be a "jump" discontinuity- that is, that the left and right limits exist but are different? You need that to be able to talk about "lim(x>x0)f(x)>lim(x<x0)f(x)". But I don't see how your idea is going to work. Your g(x) doesn't actually use "lim(x>x0)f(x)>lim(x<x0)f(x)" It just says g(x)= x if f is discontinuous at x- and assumes f is discontinous at x! You mistake is looking at the domain instead of the range.

Try this. Let [itex]lim_{x\rightarrow x_0^-} f(x)= a[/itex], [itex]lim_{x\rightarrow x_0^+} f(x)= b[/itex]. Since f has only "jump" discontinuities, those exist and a< b. There exist at least one rational number in the interval (a,b). Let g(x0) be such a rational number. Because f is an increasing function, if x0, x1 are points of discontinuities of f, x1> x0, then the "a" corresponding to x1 is greater than the "b" corresponding to x0: the two intervals do not overlap and so [itex]g(x_0)< g(x_1)[/itex]. g is a one-to-one function from A to a subset of Q.
 
  • #3
how have you defined the function g, i mean i don't see any explicit definition of the function, and how do you deal with points of discontinuity which are irrational, obviously they arent mapped into Q.
 
  • #4
All I can do is repeat what I said- If xi is a point of discontinuity of the increasing function f. Then [itex]lim_{x\rightarrow x_i^-} f(x)= a_i[/itex], [itex]lim_{x\rightarrow x_i^+} f(x)= b_i[/itex] exist and ai< bi.
Choose a rational number yi in the interval (a,b). Such a rational number certainly exists, just choose one. Define g(xi)= yi. Since, if xj> xi, as I said before, since f is increasing, bi< aj so the intervals do not overlap and [itex]y_j\ne y_i[/itex]. g is a one-to-one function from the set of points of discontinuity of f into Q. Whether xi is rational or not is irrelevant. The corresponding g(xi) is yi which is, by definition, rational.
 
  • #5
shouldn't it be:
b_i>a_j?
 
  • #6
No! That's the whole point. If [itex]a_j< b_i< b_j[/itex] then the two intervals overlap so it's possble that [itex]y_i= y_j[/itex] and the function may not be one-to-one.

Because f is increasing, if [itex]x_i< x_j[/itex], then the limit at xi from above must be less than the limit at xj from below: [itex]b_i< a_j[/itex].
 

Related to How Can Monotone Functions Have Only Countably Many Discontinuities?

1. What is a set of discontinuous points?

A set of discontinuous points is a collection of points on a graph or function that are not connected. This means that there are gaps or breaks in the graph where the points are not defined or do not exist.

2. How do you identify a set of discontinuous points?

To identify a set of discontinuous points, you must look for breaks or gaps in the graph or function. These breaks can be seen as vertical asymptotes, holes, or sharp turns in the graph.

3. What causes a set of discontinuous points?

A set of discontinuous points can be caused by a variety of factors, including undefined or discontinuous functions, non-permissible values, or discontinuities in the domain.

4. Can a set of discontinuous points be continuous at any point?

No, a set of discontinuous points cannot be continuous at any point. Continuity requires that the function be defined and have a limit at that point, which is not possible if there is a break or gap in the graph.

5. How are set of discontinuous points relevant in science?

In science, sets of discontinuous points can represent important phenomena such as sudden changes in data, phase transitions, or abrupt shifts in a system. They can also be used to model real-world scenarios, such as population growth or chemical reactions.

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