- #1
mikeph
- 1,235
- 18
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"
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"