Is y = abs (x) Continuous?

Date: 8/29/96 at 0:58:5
From: Jim Hayes
Subject: Is y = abs (x) Continuous?

Is the absolute value function (y equals the absolute value of x) a 
continuous function?

Date: 8/29/96 at 11:6:41
From: Doctor Mike
Subject: Re: Is y = abs (x) Continuous?

Hello,

Yes, f(x) = |x| = abs(x) is a continuous function. Roughly, that 
means that it has no gaps, so you can draw it's graph without lifting 
your pen and moving it. (Of course, I'm skipping over the problems of 
the pen running out of ink since the graph is infinite!)
  
If you graph it, however, you will notice that it makes a sharp turn 
as it crosses the y axis, so that the graph amounts to one giant V 
shape.  It is because of this sharp turn that f(x) does not have a 
derivative at x=0.  Everywhere else it does, though. When the 
derivative exists at every x where the function itself is defined, 
then the function is  called differentiable.  So, the absolute value 
function is continuous but not differentiable.  
  
I hope this helps.  Write back if you have more questions.  

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