Defining 'Undefined'

Date: 09/15/2003 at 21:21:13
From: Mike 
Subject: Is a hole in a graph (of a function) undefined?

I know that division by zero is undefined, and can be graphed as an 
asymptote. Is a hole in the graph (an open circle) undefined also? Or 
is it called something else like 'non-existent'? I know they are 
both discontinuities.

If a function is 'undefined at x', should I think of all 
discontinuities, or just vertical asymptotes, or are there more ways
for it to be undefined?

This isn't an important question (and it's kind of picky), so feel
free to skip it if you are busy. 

Thanks.

Date: 09/15/2003 at 23:01:23
From: Doctor Peterson
Subject: Re: Is a hole in a graph (of a function) undefined?

Hi, Mike.

Why do you draw a hole in a graph? Because the function has no value 
for that x, right? And that is what "undefined" means. The function 
is not defined for that x. 

I get the impression that many students (and teachers, too) miss the
simple meaning of "undefined", and imagine that it is some magical
state like "infinity", rather than merely meaning that something is
not defined, or does not exist. For that reason, I don't consider this
a "picky" question at all; it's very important!

Asking where a function is undefined is essentially just asking for 
the complement of its domain, which means any place it "lets light 
through" onto the x axis, hole or otherwise.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/