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Nondifferentiable Functions


Date: Sun, 6 Nov 1994 14:24:49 -0500 (EST)
From: Peter Finin
Subject: Non differentiable function.

In my calculus book, it mentions a function that is not differentiable at 
any point due to the fact that it is not smooth at any point.  It does not
go any farther than this, and I was interested in hearing more about this.

Thanx.

-Peter Finin


From: Dr. Ken
Date: Sun, 6 Nov 1994 14:53:24 -0500 (EST)

Peter!

Great question!  If what you want to see is an example of a nasty function
like that, I can give you one.  In particular, let f(x) be a function that
gives 0 as its output when x is rational, and 1 as its output when x is
irrational.

Since you're taking Calculus, I'll assume that you know about irrational and
rational numbers, particularly that between any two irrational numbers is a
rational number, and that between any two rational numbers is an irrational
number.  So the graph of this function looks kind of like a couple of blurry
lines (that's how I think of it, anyway.  It's not very technically precise,
but I think it helps get an idea of how it looks), one at height zero and
one at height one.  

Quite a jagged function.  Is it continuous anywhere?

Here's another one for you:  
F(x) = 0 when x is rational, and x when x is irrational.  What are its
properties?  Where is it continuous?  What does it look like?  Where is it
differentiable?

-Ken
    
Associated Topics:
High School Calculus
High School Functions

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