§1.7 Limits, continuity, and differentiability (continued)

Summary

A function $p$ is built from five straight-line segments with slopes $-2$, $-\frac{1}{2}$, $1$, $0$, and $3$ left to right, joined with corners, a jump at $x = 0$, and a missing value at $x = 3$. What does the graph of $y = p'(x)$ look like?

For the same $p$: the two sides at $x = 0$ approach $0$ and $-2$, the value $p(3)$ does not exist, and the graph has corner points at $x = -2$, $1$, and $2$. Where is $p$ discontinuous, and where not differentiable?

A function $h$ has a derivative $h'$ whose graph is piecewise constant with values $\pm 1$, undefined at $x = -2, 0, 2$. What can be concluded about $h$?

Let $g(x) = \sqrt{|x|}$. What happens to the difference quotient $\frac{\sqrt{|h|}}{h}$ as $h$ approaches $0$?

Which function is differentiable at $a = 3$ but has no limit at $a = 3$?

Let $h(x) = 1$ for every $x \ne -2$, with $h(-2) = 4$. What is true at $a = -2$?

For $g(x) = \sqrt{|x|}$, evaluate the difference quotient value $\frac{\sqrt{|0.01|}}{0.01}$, rounded to three decimals.

For the same $g(x) = \sqrt{|x|}$, evaluate $\frac{\sqrt{|-0.001|}}{-0.001}$, rounded to two decimals.

A function can be continuous at every point and still fail to be differentiable at some point.