Why Differentiability Implies Continuity

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

Theorem:
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,
 
BN

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 
below:

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

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/