Exercise 4.5.2

Question

Provide an example of each of the following, or explain why the request is impossible
  \begin{enumerate}[label=(\alph*)] 
  \item A continuous function defined on an open interval with range equal to a closed interval.
  \item A continuous function defined on a closed interval with range equal to an open interval.
  \item A continuous function defined on an open interval with range equal to an unbounded closed set different from $\mathbf{R}$.
  \item A continuous function defined on all of $\mathbf{R}$ with range equal to $\mathbf{Q}$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Possible, see Exercise 4.4.8 (b)
  \item Impossible by preservation of compact sets
  \item Let $f : (0,1) 	o [2, \infty)$ be defined by
    $$
    f(x) = \begin{cases}
      \frac 1x &	ext{ if } x \in (0, 1/2] \
      \frac{1}{1-x} &	ext{ if } x \in (1/2, 1)
    \end{cases}
    $$
    This works since $[2,\infty)$ is closed, unbounded and different from $\mathbf{R}$.
  \item Impossible as this contradicts the intermediate value theorem.
   \end{enumerate}

Exercise 4.5.2

Answers

Comments

Exercise 4.5.2

Answers

Comments

Add answer