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Looking at $f(x) = |x|^\alpha$ for $x \neq 0$ and $f(0)=0$. Which of these following statements is true?

a. A $\alpha$ exists such that $f \in \mathcal {L}(\mathbb {R})$.

b. A $\alpha$ exists such that $f \in \mathcal {L}([-1,1])$.

c. A $\alpha$ exists such that $f \in \mathcal {L}(\mathbb {R} \setminus [-1,1])$.

I think that a is true, that b is wrong and that c is true; is that correct?

Maria M.
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1 Answers1

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a) is false and b) and c) are true.

$\int_{-1}^{1}|x|^{\alpha} dx<\infty$ iff $\alpha >-1$

$\int_{|x|>1} |x|^{\alpha} dx <\infty$ iff $\alpha <-1$

Since these two conditions cannot both be satisfied it follows that $f$ can never be integrable on $\mathbb R$.

Kavi Rama Murthy
  • 311,013