Example of a bounded, increasing, discontinuous function

[k3k3]

Homework Statement

Define a function f:ℝ->ℝ that is increasing, bounded, and discontinuous at every integer.

Homework Equations

The Attempt at a Solution

I've tried defining a fuction using the greatest integer function but I cannot get it to be bounded with jump discontinuities.

Floor(x)*arctan(x) and variations like that are what I keep going toward.

 

[George Jones]

I cannot get it to be bounded with jump discontinuities.

Jump discontinuities don't have to be of the same size.

 

[Dick]

Homework Statement

Define a function f:ℝ->ℝ that is increasing, bounded, and discontinuous at every integer.

Homework Equations

The Attempt at a Solution

I've tried defining a fuction using the greatest integer function but I cannot get it to be bounded with jump discontinuities.

Floor(x)*arctan(x) and variations like that are what I keep going toward.

You might find it lot easier to describe your function piecewise on intervals [n,n+1] rather than trying to write a single formula for it.

 

[k3k3]

How about arctan(x) if x is in (n, n+1) and arctan(x)+1/x if x is an integer?

 

[skiller]

By what definition are you using the term "increasing"?

I've always known "increasing" to be different from "strictly increasing" but I understand some people may use the former to mean the latter.

If you don't mean "strictly increasing", then you can define a piece-wise function much more simply than using arctan or such like. (In fact, you can even if you do want it strictly increasing.)

 

[SammyS]

How about arctan(x) if x is in (n, n+1) and arctan(x)+1/x if x is an integer?

Is that increasing?

 

[Dick]

How about arctan(x) if x is in (n, n+1) and arctan(x)+1/x if x is an integer?

That's not increasing when x is an integer. Your function jumps up and then jumps back down, does't it? Define f(n)=arctan(n) where n is an integer. Now you just have to describe f on the intervals (n,n+1). How about using a linear function?

 

[STEMucator]

Define a function f:ℝ->ℝ that is increasing, bounded, and discontinuous at every integer.

Hmm try f(x) = 1/Integer(x) maybe? When you say 'bounded' do you mean your function could be bounded below or above? Then yeah it would be easy just to define a piecewise function using intervals.

Maybe even f(x) = Integer(x)/sinx would work too?

Where Integer(x) means the integer part of x.

 

[k3k3]

When you say 'bounded' do you mean your function could be bounded below or above? Then yeah it would be easy just to define a piecewise function using intervals.

By bounded I mean it is bounded above and below, yeah.

That's not increasing when x is an integer. Your function jumps up and then jumps back down, does't it? Define f(n)=arctan(n) where n is an integer. Now you just have to describe f on the intervals (n,n+1). How about using a linear function?

I can't think of a linear function that would scale with the arctan. How about 1/x when x isn't an integer?

 

[STEMucator]

By bounded I mean it is bounded above and below, yeah.




I can't think of a linear function that would scale with the arctan. How about 1/x when x isn't an integer?

Sorry I meant Integer(x) in my last post, i edited it.

 

[happysauce]

Maybe you can use the floor/ceiling function somehow and try to make the next jump up smaller.

 

[k3k3]

Maybe you can use the floor/ceiling function somehow and try to make the next jump up smaller.

I've thought about that, but I have had no luck in doing so.

 

[Dick]

I can't think of a linear function that would scale with the arctan. How about 1/x when x isn't an integer?

Define a different linear function in each integer interval [n,n+1]. It doesn't have to do much. It's values just have to lie between arctan(n) and arctan(n+1) and it has to increase and be discontinuous at at least one endpoint.

 

[k3k3]

Define a different linear function in each integer interval [n,n+1]. It doesn't have to do much. It's values just have to lie between arctan(n) and arctan(n+1) and it has to increase and be discontinuous at at least one endpoint.

So I just need a function whose values are between -pi/2 to pi/2?

 

[Dick]

So I just need a function whose values are between -pi/2 to pi/2?

No. I don't think you actually have been following me. Draw a graph of all of the points (n,arctan(n)) for n an integer. Join adjacent points with a line segment. Now decrease the slope of each line segment so it doesn't hit the upper point. That will give you the sort of graph you want, yes?

 

[Dick]

If you don't like like the explicit piecewise approach, what does arctan(floor(x)) look like? How could you modify it?