Exercise 4.6.1

Question

Using modifications of Dirichlet and Thomae's functions, construct a function $f: \mathbf{R} \rightarrow \mathbf{R}$ so that
  \begin{enumerate}[label=(\alph*)] 
  \item $D_{f}=\mathbf{Z}^{c}$.
  \item $D_{f}=\{x: 0<x \leq 1\}$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Modify the function continuous only at zero to be continuous around integers.
    $$
    f(x) = \begin{cases}
      x - \text{round}(x) &\text{if $x \in \mathbf{Q}$} \
      0 &\text{if $x \in \mathbf{I}$}
    \end{cases}
    $$
where $\text{round}(x)$ rounds $x$ to the nearest integer.

\item Modify the function continuous only at zero to be continuous below 0 and above 1. Let
    $$
    d(x) = \begin{cases}
      x &\text{if $x \in \mathbf{Q}$} \
      0   &\text{if $x \in \mathbf{I}$}
    \end{cases}
    $$
then $$
    g(x) = \begin{cases}
        0 & \text{if } x \leq 0 \
        d(x) & \text{if } 0 < x \leq 1 \
        2 & \text{if } 1 < x
    \end{cases}
$$
   \end{enumerate}

Exercise 4.6.1

Answers

Comments

Exercise 4.6.1

Answers

Comments

Add answer