# Monotone function with predescribed discontinuities

• e_to_the_i_pi
In summary, The unit jump function I(x) is defined piecewise as 0 when x<0 and 1 when x>=0. Given a countable subset {x_n} of the real numbers and a sequence of positive real numbers {c_n} with a sum of 1, the function f(x) can be defined as the sum of c_n * I(x-x_n). It can be proven that the limit of f(x) as x approaches negative infinity is 0 and the limit as x approaches infinity is 1. This can be shown by first proving the monotonicity and boundedness of f(x) on a finite interval [a,b], and then extending it to the entire real line.
e_to_the_i_pi
Let I(x) (the unit jump function) be defined piecewise by I(x)=0 iff x<0 and I(x)=1 iff x>=0.

Let {x_n} be a countable subset of R and {c_n} a sequence of positive real numbers satisfying \sum{c_n}=1. Let f:R \to R be defined by f(x)=\sum{c_n * I(x - x_n)}.

Prove that \lim_{x\to-\infty}{f(x)}=0 and \lim_{x\to\infty}{f(x)}=1.

I can prove this when {x_n} is a countable subset of (a,b) where a,b \in R. It seems as though the proof here should be very similar, but for some reason I just can't finish it..

Posting your proof for x in (a,b) may help the helpers.

Since 0 <= c_n * I(x-x_n) <= c_n, for all x \in [a,b], s_n(x) = \sum^n_{k=1}{c_k * I(x-x_k)} <= \sum^n_{k=1}{c_k} <= \sum^\infty_{k=1}c_k.

So for x \in [a,b], {s_n(x)} is monotone increasing and bounded above, thus convergent.

Since I(x-x_n) <= I(y-x_n), for all x<y, f is monotone increasing on [a,b].

Now, since x_n > a for all n, I(a-x_n)=0 for all n. Thus, f(a)=0. Also, I(b-x_n)=1 for all n, so f(b)=\sum{c_k}=1.

## 1. What is a monotone function with predescribed discontinuities?

A monotone function with predescribed discontinuities is a mathematical function that is either strictly increasing or strictly decreasing and has predetermined points of discontinuity, where the function is not continuous. This means that the function has abrupt changes in its values at certain points, rather than a smooth transition.

## 2. How is a monotone function with predescribed discontinuities different from a regular monotone function?

A regular monotone function is continuous, meaning that it has no sudden jumps or breaks in its values. However, a monotone function with predescribed discontinuities has predetermined points of discontinuity, making it non-continuous. This makes it a more specialized type of monotone function.

## 3. What are some common examples of monotone functions with predescribed discontinuities?

Some common examples of monotone functions with predescribed discontinuities include the step function, the greatest integer function, and the Heaviside function. These functions are commonly used in mathematical analysis and in engineering applications.

## 4. How can a monotone function with predescribed discontinuities be graphed?

A monotone function with predescribed discontinuities can be graphed by plotting the points of discontinuity on the graph and connecting them with a dotted line. The rest of the function can be graphed as a regular monotone function, with a solid line connecting the points.

## 5. What is the significance of studying monotone functions with predescribed discontinuities?

Studying monotone functions with predescribed discontinuities allows us to better understand the behavior of these functions and how they can model real-life situations. It also helps in solving problems related to optimization and finding the maximum or minimum values of a function.

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