We first met the Cantor set $C$ in Section 3.1. We have since learned that $C$ is a compact, uncountable subset of the interval $[0, 1]$. Define \[h(x) = \begin{cases} 1 & \text{if } x \in C \\ 0 & \text{if }x \notin C \end{cases}\] \begin{enumerate}[label=(\alph*)] \item Show $h$ has discontinuities at each point of $C$ and is continuous at every point of the complement of $C$. Thus, $h$ is not continuous on an uncountably infinite set. \item Now prove that $h$ is integrable on $[0,1]$. \end{enumerate}

Exercise 7.6.2

We first met the Cantor set $C$ in Section 3.1. We have since learned that $C$ is a compact, uncountable subset of the interval $[0, 1]$ .

Define

h (x) = {\begin{matrix} 1 & if x \in C \\ 0 & if x \notin C \end{matrix}

(a): Show $h$ has discontinuities at each point of $C$ and is continuous at every point of the complement of $C$ . Thus, $h$ is not continuous on an uncountably infinite set.
(b): Now prove that $h$ is integrable on $[0, 1]$ .

See Exercise 7.3.9 (d)

ulissemini

2022-01-27 00:00