Can a function on 2D be piecewise continuous?

[mikeph]

I have a definition that a piecewise continuous function is one which is continuous on all but a finite number of points. I believe a step function would be a good example.

However in 2D space, an equivalent function to the step function (eg, for x>0, y>0, f(x,y) = 1 else f(x,y) = 0) does not satisfy this definition because there are an infinite number of points where this is not continuous. Surely this step function is a reasonable 2D extension.

It seems that either "piecewise" is purely intended for 1D functions, or the mathworld definition is discriminating against higher dimensions! In either case, what can I call my "2D step function"? I simply want it to mean a function which can have 2D steps, but nothing more than that.

Thanks for any help.

PS. Mathworld definition:
"A function or curve is piecewise continuous if it is continuous on all but a finite number of points"

 

[Grufey]

Hello MikeyW.

I'm not sure, but, sometimes in functional analisys, we say that, the function is continuous, except in sets of null measure. In this case, the function f(x,y)=1 if x>0, y>0, else f(x,y)=0, is continuous, except in sets of null measure.