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### Why Differentiability Implies Continuity

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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
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```
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/
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Associated Topics:
High School Calculus

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