demonelite123
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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 a little stuck on part b). i am trying to use the dominated convergence theorem but i am a bit confused on what function is dominating what. since F is L-integrable it can be written as an infinite sum of L integrable functions but it is not clear to me how i can use that to show that the limit as x approaches +/- infinite of F(x) = 0. is there a better way to think of F(x) that can help me prove this fact?
b) show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.
i am a little stuck on part b). i am trying to use the dominated convergence theorem but i am a bit confused on what function is dominating what. since F is L-integrable it can be written as an infinite sum of L integrable functions but it is not clear to me how i can use that to show that the limit as x approaches +/- infinite of F(x) = 0. is there a better way to think of F(x) that can help me prove this fact?