Calculus 1 question

[Steven G]

Can anyone think of a non-piecewise function that has a jump discontinuity?

 

[Cubist]

The Fourier series of a piecewise function that has a jump discontinuity.

I anticipate your next question, "Can anyone think of a non-piecewise, and non-infinite sum function that has a jump discontinuity?"

 

[Dr.Peterson]

Can anyone think of a non-piecewise function that has a jump discontinuity?

How about the floor function?

 

[Steven G]

How about the floor function?

I would say no, as it is defined as a piecewise function. I too thought of that but rejected it. For the record I have a very nice function. Nothing strange or fancy just a very clean looking function.

Should I post it now or does anyone want more time?

 

[Dr.Peterson]

I would say no, as it is defined as a piecewise function. I too thought of that but rejected it. For the record I have a very nice function. Nothing strange or fancy just a very clean looking function.

Should I post it now or does anyone want more time?

Actually, have you tried formally defining the floor function piecewise? (It has infinitely many pieces!) I agree that it seems inherently piecewise; but I imagine that could be said about anything with a jump.

The inverse secant is inherently "in pieces", and I imagine you might be able to construct a jump discontinuity from that. In fact, one version of the inverse cotangent in itself has a jump, by virtue of the choice of range:

 

[LCKurtz]

The signum function sgn(x). Or [MATH]|x|'[/MATH]. Or do they violate the spirit of the problem?

 

[Steven G]

f(x) = e^(1/x)/[1+e^(1/x)]

This will be my favorite function for a while.

https://www.wolframalpha.com/input/?i=graph+f(x)+=+e^(1/x)/[1+e^(1/x)]

 

[Dr.Peterson]

f(x) = e^(1/x)/[1+e^(1/x)]

This will be my favorite function for a while.

https://www.wolframalpha.com/input/?i=graph+f(x)+=+e^(1/x)/[1+e^(1/x)]

But that's undefined at x=0; I could be wrong, but I think of a jump discontinuity as being like all the other examples, which are defined at that point.

For example, MathWorld implies this:

A real-valued univariate function f=f(x) has a jump discontinuity at a point x_0 in its domain provided that ...​

 

[Steven G]

I was taught that a jump discontinuity was a nonremovable discontinuity. That is a function where you could not redefined a finite number of points to make it continuous.

 

[LCKurtz]

[MATH]\frac x {\sqrt{x^2}}[/MATH]

 

[apple2357]

does arctan(1/x) fit?

graph f(x) = arctan(1/x) - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

 

[Dr.Peterson]

[MATH]\frac x {\sqrt{x^2}}[/MATH]

does arctan(1/x) fit?

graph f(x) = arctan(1/x) - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

These both share the difficulty that they are not defined at x=0; if you allow a jump discontinuity to be like that, then it's a lot easier to find examples.

My suggestion in #5 is identical to #11 but with a hole filled in (and that's the reason for that choice of range for arccot).

 

[LCKurtz]

Just to throw in a bit of uncertainty. I'm not convinced that it is a standard definition that a piecewise continuous function on an interval must be defined at each point of the interval. Look here for an example of that:

http://websupport1.citytech.cuny.edu/faculty/vgitman/diffeqs/lecture17.pdf

I don't know where to look for an "official" definition which is generally agreed.

 

[Dr.Peterson]

Just to throw in a bit of uncertainty. I'm not convinced that it is a standard definition that a piecewise continuous function on an interval must be defined at each point of the interval. Look here for an example of that:

http://websupport1.citytech.cuny.edu/faculty/vgitman/diffeqs/lecture17.pdf

I don't know where to look for an "official" definition which is generally agreed.

I never said I have no uncertainty about this!

But to be technical, the question is not about the definition of "piecewise continuous", but of "jump discontinuity" in a "non-piecewise" function (that is, not defined by pieces); and the initial question implied that what was asked for is rare. That subconsciously, at least, led me to ignore the relatively common situation like x/|x|, and consider only functions that are discontinuous only because of a jump, and not (also) because of a hole. Then I found one definition that agrees with that impression.

There are lot of definitions (especially of relatively informal terms like "jump discontinuity") that are not quite clear.