The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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.


-Peter Finin

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


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

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

Associated Topics:
High School Calculus
High School Functions

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.