Exercise 4.4.8

Question

Give an example of each of the following, or provide a short argument for why the request is impossible.
  \begin{enumerate}[label=(\alph*)] 
  \item A continuous function defined on $[0,1]$ with range $(0,1)$.
  \item A continuous function defined on $(0,1)$ with range $[0,1]$.
  \item A continuous function defined on $(0,1]$ with range $(0,1)$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Impossible as continuous functions preserve compact sets and $(0,1)$ is not compact.
  \item Define
    $$
    f(x) = \begin{cases}
      3x - 1 &\text{ if } x \in [1/3, 2/3] \
      1      &\text{ if } x \in (2/3, 1) \
      0      &\text{ if } x \in (0, 1/3) \
    \end{cases}
    $$
    $f$ is continuous on $(0,1)$ and has range $[0,1]$. Here's a graph of $f$:
    \begin{tikzpicture}
      \draw (0, 0)   -- (1/3, 0);
      \draw (1/3, 0) -- (2/3, 1);
      \draw (2/3, 1) -- (1, 1);
    \end{tikzpicture}
  \item Consider $g(x) = \sin(1/x) (1-x)$ over $x \in (0, 1]$; clearly $g(x)$  is continuous over this interval. The $1-x$ term bounds $g(x)$ to $(-1, 1)$, while the $\sin(1/x)$ ensures that $g(x)$ will approach this bound arbitrarily close as $x \to 0$. Thus, the range of $g(x)$ is $(-1, 1)$.
  We now just need to shape this to $(0, 1)$ by defining $f(x) = (g(x) + 1) / 2$.
   \end{enumerate}

Exercise 4.4.8

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Exercise 4.4.8

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