Continuity

Date: 9/1/96 at 19:18:56
From: Jim Hayes
Subject: Continuity

Hello Dr.Math,

My teacher friend states that the absolute value function is not 
continuous.  His reasoning is based on a definition he uses in 
secondary school: 

  "A discontinuous function is a function specified by different rules 
   over different values for its domain."  

As an example, when looking at the graph of the absolute value 
function, he says this graph could also be the graph of the union of 
two functions, f(x)=x, where the domain is restricted to values equal 
to or greater than zero, and g(x)=-x, where the domain is restricted 
to values equal to or less than zero.  

Of course the graphs are identical, but I believe that his definition 
is all wet.  What his definition leads to is that any graph that 
changes direction in a manner which is not smooth, is discontinuous.  

What do think of his reasoning?

Jim Hayes

Date: 9/2/96 at 12:39:22
From: Doctor Ken
Subject: Re: Continuity

Hello!

You are absolutely correct.  The absolute value function is definitely 
continuous.  The "different rules for different parts" criterion might 
work in some cases (or rather, it can give you clues for which 
functions you should look at more closely), but it's not the 
definition of continuity (for more information on continuity, you can 
browse our archives).

Note further that you can define the absolute value function a 
different way: |x| = Sqrt(x^2).  This produces the exact same 
function, so if one function is continuous then the other one is.  
Using the "different rules for different parts" criterion would give 
you different results for these two ways of defining, so something's 
fishy.

A better way of thinking about continuity (although it too is not the 
definition) is whether you can draw the function without picking up 
your pencil.

The actual definition?  

Given x, the limit as c approaches x exists and it's equal to f(x)
if and only if f is continuous at x.

Then you say f is continuous (all over) if it's continuous at every x.

Hope this helps.

Oh, by the way, what the absolute value function is _not_ is 
differentiable.  That means it's got a sharp edge.

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/