Continuity and differentiability of $x^a$ and $x^a\sin(1/x)$ at 0

1. Let

\[ f _ { a } ( x ) = { \left\{ \begin{array} { l l } { x ^ { a } } & { { \mathrm { i f ~ } } x > 0 } \\ { 0 } & { { \mathrm { i f ~ } } x \leq 0 } \end{array} \right. } \]

1. For which values of a is \(f _ { a }\) continuous at \(0 ?\)

2. For which values of a is \(f _ { a }\) differentiable at 0? In this case is the derivative function continuous?

3. For which values of a is \(f _ { a }\) twice-differentiable?

2. Let

\[ g _ { a } ( x ) = { \left\{ \begin{array} { l l } { x ^ { a } \sin ( 1 / x ) } & { { \mathrm { i f ~ } } x \neq 0 } \\ { 0 } & { { \mathrm { i f ~ } } x = 0 } \end{array} \right. } \]

Find particular non-negative (and potentially non-integral) values of a for which:

1. \(g _ { a }\) is differentiable on \(\mathbb { R } ,\) but \(g _ { a } ^ { \prime }\) is unbounded on [0, 1].

2. \(g _ { a }\) is differentiable on R with \(g _ { a } ^ { \prime }\) continuous but not differentiable at 0.

3. \(g _ { a }\) and \(g _ { a } ^ { \prime }\) are differentiable on R, but \(g _ { a } ^ { \prime \prime }\) is not continuous at 0.