Can the value of a function at a point of discontinuity vary among authors?

[spaghetti3451]

In their definitions, the value of the function at the point of discontinuity varies from one author to another. Why is this the case?

Also, the authors draw a vertical line at the point of discontinuity. Isn't this incorrect? (surely, the function cannot be drawn as multivalued if it is single-valued in the definition?)

I am also wondering if the function can have multiple values at the point of discontinuity.

Thanks in advance!

 

[micromass]

In their definitions, the value of the function at the point of discontinuity varies from one author to another. Why is this the case?

The reason that we want to study step functions is because we're interested in the surfac area beneath the function. But I think it's easy to see that the surface area beneath a finite number of points is always zero. So if you change a finite number of points in a function, then the surface area will not change. So what value we take doesn't really matter, and this is the reason that it's allowed for different authors to define the value at the point of discontinuity differently.

Also, the authors draw a vertical line at the point of discontinuity. Isn't this incorrect? (surely, the function cannot be drawn as multivalued if it is single-valued in the definition?)

No, this is not correct. Can you give me a reference to an author that actually does this, because I don't think I've ever seen such a thing.

I think that the problem is with the graphing software. Often it can not draw discontinuities, but it can only draw a vertical line. But this is a mistake, the vertical line shouldn't be there.

I am also wondering if the function can have multiple values at the point of discontinuity.

No, this is not allowed.
What some authors do however, is not to define the function at the point of discontinuity, since it doesn't matter. But I also don't consider this to be very rigorous and I don't like it...

 

[lurflurf]

The definition of f(a) varies, somtimes it is undefined, somtimes something like (f(a-)+f(a+))/2 because that is what Fourier series give. Sometimes he funcion is made continuous by smothing then the line is not vertical it only looks vertical. Even when the discontinuity is kept the vertical like is not incorrect, but it does not belong to the function, it is just to show the discontinuity. In many cases the actual value is unimportant.

 

[spaghetti3451]

But I think it's easy to see that the surface area beneath a finite number of points is always zero. So if you change a finite number of points in a function, then the surface area will not change.

Shouldn't there at least be some finite non-zero area beneath the finite number of points even if it might be negligibly small? I ask this becuase you say that the area under those points is zero.

No, this is not correct. Can you give me a reference to an author that actually does this, because I don't think I've ever seen such a thing.

In drawing a square wave, for instance. See http://en.wikipedia.org/wiki/Square_wave. I agree this is not a textbook digram, but then again every teacher draws a vertical line at the point od discontinuity. What should I believe?

 

[micromass]

Shouldn't there at least be some finite non-zero area beneath the finite number of points even if it might be negligibly small? I ask this becuase you say that the area under those points is zero.

No, the area under a finite number of points is always zero. This is not just true in integrals, but it already follows from classical geometry: imagine a rectangle with two sides, one side is 5 meters long, the other is 0 meters long. Then the area of the rectangle is 0x5 meters². Thus the area of the rectangle is 0m².

The tricky question is: how do you define the concept of area? This is a very interesting questions with a number of answers. But every answer will agree that the area under a point is zero. I know this may sound counterintuitive, but it's something that needs to be like that.
The point is that these diagrams are mostly used by engineers. And I don't mean to say anything bad about engineers, but their notation and rigor may be a bit sloppy at times...

In drawing a square wave, for instance. See http://en.wikipedia.org/wiki/Square_wave. I agree this is not a textbook digram, but then again every teacher draws a vertical line at the point od discontinuity. What should I believe?

I see... Well, I've seen these diagrams before. But they're actually "wrong": the straight lines don't need to be there. However, many authors already use these diagrams, so nobody will change it. You simply need to remember that the straight lines are not really there...

 

[spaghetti3451]

Thanks!