Exercise 6.5

Question

Exercise 5: 
    Suppose $f$ is a bounded real function on $[a,b]$, and $f^2\in\mathscr R$ on $[a,b]$. Does it follow that $f\in\mathscr R$? Does the answer change if we assume that
    $f^3\in\mathscr R$?

ghostofgarborg · Accepted Answer

(	extit{Matt ``Frito'' Lundy}) \
    To answer the question ``Does $f^2 \in \mathscr R$ imply that $f \in \mathscr R$?'' consider the function
    $$f(x) = 
    \begin{cases}
        1 & x 	ext{ rational} \
        -1 & x 	ext{ irrational}.
    \end{cases} $$
    Then $f^2 = 1$ and $f^2 \in \mathscr R$, but $f 
otin \mathscr R$.

To answer the question ``Does $f^3 \in \mathscr R$ imply that $f \in \mathscr R$?'' 
    use the fact that $f$ is bounded on $[a,b]$ implies that $f^3$ is bounded on $[a,b]$,
    so there exists $m, M \in \mathbf R$ such that $m\leq f^3 \leq M$ for all $x \in [a,b]$.
    Let $\phi (x) = x^{1/3}$, then $\phi$ is continuous on $[m,M]$, and $f = \phi (f^3)$,
    so $f \in \mathscr R$ on $[a,b]$ by theorem 6.11.

Notice that $\phi(x) = x^{1/2}$ does not work for the first question because $\phi$ is not continuous on $[-1,1]$.

Exercise 6.5

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Exercise 6.5

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