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### 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/
```
Associated Topics:
High School Functions

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