Exercise 4.4.12

Question

Review Exercise 4.4.11, and then determine which of the following statements is true about a continuous function defined on $\mathbf{R}$:
  \begin{enumerate}[label=(\alph*)] 
  \item $f^{-1}(B)$ is finite whenever $B$ is finite.
  \item $f^{-1}(K)$ is compact whenever $K$ is compact.
  \item $f^{-1}(A)$ is bounded whenever $A$ is bounded.
  \item $f^{-1}(F)$ is closed whenever $F$ is closed.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item False, $f(x)=0$ has $f^{-1}(\{0\}) = \mathbf R$
  \item False, $f(x)=0$ has $\{0\}$ compact but $f^{-1}(\{0\}) = \mathbf{R}$ is not compact
  \item False, $f(x)=0$ has $f^{-1}(\{0\}) = \mathbf{R}$
  \item True, let $(x_n) 	o x$ be a convergent sequence in $f^{-1}(F)$, we know $f(x_n) 	o f(x) \in F$ meaning $x \in f^{-1}(F)$ and so $f^{-1}(F)$ is closed.
   \end{enumerate}

Exercise 4.4.12

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Exercise 4.4.12

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