Finiteness of the integral of the square of a non-negative continuous unbounded function

Question

Let $f:[0,\infty)\to [0,\infty)$ be a continuous function with $\int_{0}^{\infty}f(x)dx<\infty$. Can we conclude that $\int_{0}^{\infty}(f(x))^2dx<\infty$? I understand that the conclusion is true if $f$ is bounded by some positive $M$ because then $\int_{0}^{\infty}f^2<M\int_{0}^{\infty}f<\infty$. But I wonder if there is a counterexample when $f$ is unbounded.

Any help or comment is appreciated. Thanks in advance.

Answer 1

Hint: $f=\sum n1_{(n,n+n^{-3})}$ has these integrability properties. modify this to get a continuous function.