# Need some help in showing that the function f '(x) is not continuous at 0.

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• Mar 4th 2013, 05:59 AM
marsalanuddin
Need some help in showing that the function f '(x) is not continuous at 0.
Here is the problem: Consider the function f(x) defined by
f(x)= x2sin(1/x) if x does not = 0
0 if x = 0

Show that the function f ' (x) is not continuous at 0.

Any help would be appreciated.
• Mar 4th 2013, 07:12 AM
TheEmptySet
Re: Need some help in showing that the function f '(x) is not continuous at 0.
Quote:

Originally Posted by marsalanuddin
Here is the problem: Consider the function f(x) defined by
f(x)= x2sin(1/x) if x does not = 0
0 if x = 0

Show that the function f ' (x) is not continuous at 0.

Any help would be appreciated.

Use the limit definition of the derivative and the squeeze theorem to prove that $\displaystyle f(x)$ is differentiable. Then just use the product and chain rule to find the derivative. Once you have $\displaystyle f'(x)$ show that $\displaystyle \lim_{x \to 0}f'(x) \ne f'(0)$

Hint the limit of the derivative does not exist