Exercise 5.13

First note that $\sin <!--\; FUNCTION\; APPLICATION\; -->\left(|x{\text{|}}^{-c}\right)$ fluctuates between $-1$ and 1, and each neighborhood of 0 has an infinite number of elements from each of the sets $f^{-1}(0)$, $f^{-1}(1)$ and $f^{-1}(-1)$.

(a) $f$ is continuous at 0 if and only if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}f(x)=0$ which only happens if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}|x{\text{|}}^a=0$, that is $a>0$.

(b) $f^{\prime }(0)=\underset{t\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}f(t)∕t=\underset{t\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}{t}^{a-1}\sin <!--\; FUNCTION\; APPLICATION\; -->\left(|t{\text{|}}^{-c}\right)$ exists if and only if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}|x{\text{|}}^{a-1}$ exists, that is $a>1$.

(c) For $x>0$,

which is bounded on $(0,1]$ if and only if $a-c-1>0$, that is $a>1+c$. By symmetry, the same is true on $[-1,0)$.

(d) $f^{\prime }$ is continuous at 0 if and only if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}^{\prime }(x)=0$ which only happens if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}|x{\text{|}}^{a-c-1}=0$, that is $a>1+c$.

(e) $\underset{t\to 0+}{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}^{\prime }(t)∕t=\underset{t\to 0+}{\lim <!--\; FUNCTION\; APPLICATION\; -->}\left(a{t}^{a-2}\sin <!--\; FUNCTION\; APPLICATION\; -->(t^{-c})-c{t}^{a-c-2}\cos <!--\; FUNCTION\; APPLICATION\; -->(t^{-c})\right)$ exists and is equal to 0 if and only if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}|x{\text{|}}^{a-c-2}$ exists, that is $a>2+c$. We have the same result when taking the limit from the left.

(f) For $x>0$,

which is bounded on $(0,1]$ if and only if $a-2c-2>0$, that is $a>2+2c$. By symmetry, the same is true on $[-1,0)$.

(g) $f^{″}$ is continuous at 0 if and only if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}^{″<!--\; FUNCTION\; APPLICATION\; -->}(x)=0$ which only happens if $\underset{x\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}|x{\text{|}}^{a-2c-2}=0$, that is $a>2+2c$.