lebesgue integrability and showing limit as x approaches +/-∞ of F(x) = 0.

a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous.

b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.

i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help.