Interesting Continuity Question

Date: 09/01/2005 at 18:43:05
From: Maggie
Subject: Calculus Continuity in Functions

Is every function continuous at some point in its domain?

I thought that every function didn't have to have a continuous point, 
therefore not every function would be continuous at some point in its 
domain, but I'm not sure.

Date: 09/02/2005 at 19:10:45
From: Doctor Julien
Subject: Re: Calculus Contiunity in Functions

Hi Maggie,

Thanks for writing!  No, not every function has to have a point in its
domain where it is continuous.  Formal definition aside, continous
means the function is "smooth".

Take the following function:

  f(x) = 1 if x is rational
        -1 if x is irrational

This function "jumps around" from 1 to -1, and has no points that are
continous anywhere!

You can see a picture of a similar function here:

  DirichletFunction
    http://mathworld.wolfram.com/DirichletFunction.html 

Does this help?

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