Drexel dragonThe Math ForumDonate to the Math Forum

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

Why Differentiability Implies Continuity

Date: 03/06/98 at 19:51:48
From: bill noorduyn
Subject: Calculus: differentiability

If a function is differentiable in an interval then it must be 
continuous in that interval.

Question:  Why?  The basic definition of the derivative is as a 
special limit. In evaluating the special limit, say, as x approaches 
a, we need never consider the value x = a. So a limit can exist in an 
interval even though we have a point of discontinuity at the point 
(a, f(a)). Thus I conclude that if a function is differentiable 
everywhere then it CAN be discontinuous at(many)points. It is clear 
to me that if a function is continuous everywhere it need not be 
differentiable everywhere. For example y = the absolute value of x.

Could you please concur with me or correct the error if you have the 
interest and the time.
Sincerely curious,

Date: 03/06/98 at 23:51:50
From: Doctor Wolf
Subject: Re: Calculus:  differentiability

Hi Bill,

You are absolutely right in your statement that continuity of a 
function does not imply differentiability. The absolute value function 
is a good example, and based on this "kink in the graph" problem, 
functions have been devised that are continuous everywhere, yet 
differentiable nowhere.

However, it is a basic theorem of calculus that differentiabiity at a
point implies continuity at that point. If you examine the limit 

            lim    f(a+h) - f(a)  
     f'(a)= h->0   ------------- .  

the existence of this limit would mean that f(a) not only exists, but 
the limit of f(x) as x approaches a equals f(a).

Therefore, a function which is differentiable everywhere is also 
continuous everywhere.

Excellent question ....

-Doctor Wolf,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
Associated Topics:
High School Calculus

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.