math/language question

[VJ]

Hello,

Probabely a simple question for the English speaking mathematicians here. Suppose you have a function:

f: A -> B
   x -> x^2 for x in [0,1]
   x -> x^3 for x in ]1,2]

How do you call a function like this (i.e. a function that cannot be defined using a single expression, but is defined over different sections)? Is it just called piecewise, or is there a specific term for it?

Thanks!


Jörg

[GuchiGuh]

Correct, its a "piecewise defined function"


btw, Joerg, one of your square brackets is facing the wrong way

[VJ]

Thanks, I thought there would have been a specific term for it...

While this function was merely in illustration, the bracket is NOT facing the wrong way: it is indicating a half-open interval. For this function, it wouldn't matter, but it changes if you consider this one:

f: A -> B
   x -> x^2 for x in [0,2]
   x -> x^3 for x in ]2,4]

The function is discontinous in 2, if the second piece would be defined over [2,4], you'd get 2 functionvalues for one x value (4 for the first piece and 8 for the second piece). In this situation, you'll loose a number of properties. By defining it over a half-open interval, the function is an image (I hope 'image' is the right term).


Jörg

[Umfriend]

Uhm, not that I would know about these things, but even in the first function, AFAICS, it does matter as the funtion would have two derivatives for x=2?

[VJ]

Excellent remark!
I hadn't considered that...

But I'm not convinced it would be such a problem. The function is continous but not smooth (i.e. derivative is not continous). Even in the case with the half-open interval, you would have to consider the left and right derivative (using a limit) of 2.



Jörg

[Umfriend]

Can't you limit the use of a limit ( ) to the first piece though?

[VJ]

Well yes, you only need the limit for the second piece...
(my sentence did give the wrong impression... )


Jörg

[Umfriend]

No Smurf knows math better than I do (nope, Math Smurf ain't born yet)! That's why we live in mushroowms and our dam always breaks,

[dshumake]

This type of function is also known as a step function.

Another nomenclature is using a parenthesis to indicate that the number enclosed is actually a limit to that number.

f: A -> B
x -> x^2 for x in [0,2]
x -> x^3 for x in (2,4]


There can be many steps.

dshumake

[VJ]

Correct me if I'm wrong, but it is only a step-function if it isn't continuous.

[Wombat]

This is NOT a step function.

http://en.wikipedia.org/wiki/Step_function

[Gurm]

HAHA! Between Wikipedia and Snopes... we can correct anyone, anytime!