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**shadow_2145** (1) Let $\displaystyle g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right. $

(a) For which of the values of $\displaystyle a$ is $\displaystyle f$ continuous at zero?

(b) For which values of $\displaystyle a$ is $\displaystyle f$ differentiable at zero? In this case, is the derivative function continuous?

(c) For which values of $\displaystyle a$ is $\displaystyle f$ twice-differentiable?

(2) Let $\displaystyle f$ and $\displaystyle g$ be functions defined on an interval $\displaystyle A$, and assume both are differentiable at some point $\displaystyle c \in A.$

Prove the following:

(a) $\displaystyle (f+g)'(c) = f'(c) + g'(c)$

(b) $\displaystyle (kf)'(c) = kf'(c),$ for all $\displaystyle k \in R$ (R is the reals)

Any help would be greatly appreciated, thanks!