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)

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- Apr 13th 2010, 02:06 PMsw2010If 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, 03:21 PMsw2010
please help me give me a hint

- Apr 15th 2010, 03:51 PMDrexel28
- Apr 15th 2010, 04:14 PMsw2010
if f is differentiable, then f is continuous.

- Apr 15th 2010, 04:55 PMPlato
- Apr 15th 2010, 06:10 PMsw2010
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, 06:28 PMmabruka
so how does that help? ??

- Apr 15th 2010, 06:31 PMsw2010
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.