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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.

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Solution Summary

Differentiables and continuous functions are clearly exhibited.

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

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