TITLE: Is every integrable function on the real line with compact support also square integrable?
QUESTION [7 upvotes]: 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.

REPLY [8 votes]: Consider $x \mapsto 1/ \sqrt x$