prove or disprove about piecewise continuous‏ functions

Question

just some proves or disproves, I can really use some help/clues with:

1. If F is piecewise continuous‏ in $ [-\pi,\pi]$ then it belongs to $L_2[-\pi,\pi]$

• I don't think its true, maybe $cot(x)$ disprove it? not sure.

2. If F belongs to $L_2[-\pi,\pi]$ then F is piecewise continuous‏.

• I think its true but don't know how to prove it in general.

3. If f is continuous‏ in $R$ then f belongs to $L_1(R)$

thank you!

Answer 1

1. False, take any function with a vertical asymptote that diverges sufficiently quickly

2. False, take any measurable bounded non-piecewise continuous function on $[-\pi, \pi]$, e.g. the characteristic function of the irrationals in $[-\pi, \pi]$ or the unit fraction indicator function.

3. False, take any non-null constant function.