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/
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