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I wonder that whether every integrable function on the real line with compact support is also square integrable ? In other words, is $L^1_c(\mathbb R)\subseteq L^2(\mathbb R)$ holds true? Thanks in advance for any hints.

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If $f\in L^2[a,b]$ for some $a,b\in\mathbb R$, then $f\in L^1[a,b]$. ${}\qquad{}$ – Michael Hardy Apr 27 '14 at 3:04

Consider $x \mapsto 1/ \sqrt x$

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or any function $x^a$ for $-1\lt a\le-\frac12$. (+1) – robjohn Apr 27 '14 at 9:39
How does this function have compact support? – user2357112 Apr 27 '14 at 11:02
@user2357112 I left the details to the OP. – Pedro Tamaroff Apr 27 '14 at 12:45

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