\begin{enumerate}[label=(\alph*)] \item Find $g'(0)$. \item Use the standard rules of differentiation to compute $g'(x)$ for $x \neq 0$. \item Explain why, for every $\delta > 0$, $g'(x)$ attains every value between $1$ and $-1$ as $x$ ranges over the set $-\delta, \delta$. Conclude that $g'$ is not continuous at $x = 0$. \end{enumerate}

Understanding Analysis

Exercise 7.6.14

(a): Find $g^{'} (0)$ .
(b): Use the standard rules of differentiation to compute $g^{'} (x)$ for $x \neq 0$ .
(c): Explain why, for every $δ > 0$ , $g^{'} (x)$ attains every value between $1$ and $- 1$ as $x$ ranges over the set $- δ, δ$ . Conclude that $g^{'}$ is not continuous at $x = 0$ .

See discussion in Section 5.1.

ulissemini

2022-01-27 00:00