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# Differentiable and continuous functions

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Suppose fâ?¶Râ?'R is twice differentiable with both f' and f'' continuous in an interval around 0. Suppose further that f(0)=0. Let
h(x)={f(x)/x, if xâ? 0,
f^' (0), if x=0.
Show that
(a) h is differentiable at x=0.
(b) h is differentiable at x=0 with h^' (0)=1/2 f^'' (0).
(c) h' is continuous at x=0.

https://brainmass.com/math/graphs-and-functions/differentiable-continuous-functions-395200

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Proof:
Since is twice differentiable function, and are continuous in the neighborhood of 0, and , then we have
, where is in the neighborhood of 0.
Then we get

Since and are continuous, we have

Now we consider if and .
(a) First, I claim that is differentiable at . We have

Thus is differentiable at .
(b) From (a), we get
(c) Next, I claim that is continuous at
For , we have , then . Then we have

Let , then , then we have

Therefore, is continuous at .

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