Piecewise continuous -> NO vertical asymptotes

[AxiomOfChoice]

Piecewise continuous --> NO vertical asymptotes

Why is it impossible for a piecewise continuous function to have vertical asymptotes, per this website: http://www.mathwords.com/p/piecewise_continuous_function.htm?

 

[fourier jr]

it's not impossible. you link says a piecewise-continuous function may not have vertical asymptotes, and it gives an example of one that doesn't

 

[AxiomOfChoice]

it's not impossible. you link says a piecewise-continuous function may not have vertical asymptotes, and it gives an example of one that doesn't

But it then goes on to say that "the only possible types of discontinuities for a piecewise continuous function are removable and step discontinuities." So it seems that vertical asymptotes ARE impossible!

 

[masterjai423]

"A function made up of a finite number of continuous pieces..."

It is not possible since anyone of these continuous pieces cannot have a single asymptote. Continuous functions that define these pieces are continuous in that they don't have asymptotes.

 

[LCKurtz]

The function y = 1/x for x not zero has a vertical asymptote at x = 0. I don't think you will find any text that will say otherwise. The picture on that website shows a function that is defined for all x. I think what they are trying to say is that a piecewise continuous function that is defined for all x can have no vertical asymptote. Hypotheses matter.

 

[evaniax]

It's simply part of the definition.

If you simply defined "piecewise continuous" to mean that a function has finitely many discontinuities, then a function like f(x)=1/x would satisfy that criterion (whether the function exists at x=0, is actually somewhat irrelevant in this case as we could define f(0)=0 for example). And this would give you a piecewise continuous function with a vertical asymptote.

However, most (useful) definitions of piecewise continuity involve other conditions to restrict the function from having asymptotes. If the function is continuous at all but finitely many points, each discontinuity must be isolated, meaning that at a particular discontinuous point x, the right hand and left hand limits of f exist. A typical added restriction is for all right and left hand limits to be finite for all points.