# If I is an open interval, f is differentiable on I ....

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• Apr 13th 2010, 01:06 PM
sw2010
If I is an open interval, f is differentiable on I ....
If I is an open interval, f is differentiable on I and a is in I, then there is a sequence a_n in I\{a} such that a_n-> a and f '(a_n)->f '(a)
• Apr 15th 2010, 02:21 PM
sw2010
please help me give me a hint
• Apr 15th 2010, 02:51 PM
Drexel28
Quote:

Originally Posted by sw2010
please help me give me a hint

Work. Let's see some.
• Apr 15th 2010, 03:14 PM
sw2010
if f is differentiable, then f is continuous.
• Apr 15th 2010, 03:55 PM
Plato
Quote:

Originally Posted by sw2010
if f is differentiable, then f is continuous.

Good grief. Come on. Where is any real effort on your part?
Do you understand anything about this question?
If so you do, give us something more than that!
Think about the mean value theorem.
• Apr 15th 2010, 05:10 PM
sw2010
we need to assume that f is a continuous, real-valued function, defined on an interval. If the derivative of f at every interior point of the interval I exists and is zero, then f is constant.
Proof: Assume the derivative of f at every interior point of the interval I exists and is zero. Let (a, b) be an arbitrary open interval in I. By the mean value theorem, there exists a point c in (a,b) such that 0=f '(c)=f(b)-f(a)/(b-a)

So f(a) = f(b). Thus, f is constant on the interior of I and thus is constant on I by continuity
• Apr 15th 2010, 05:28 PM
mabruka
so how does that help? ??
• Apr 15th 2010, 05:31 PM
sw2010
If I is an open interval, f is differentiable on I and a is in I, then there is a sequence a_n in I\{a} such that a_n-> a and f '(a_n)->f '(a)

if f is differentiable on an interval then it is continuous on that interval. the mean value theorem says that there is a c in the the interval.