$ (\Rightarrow) $ If there is a sequence of odd polynomials $ {p_n(x)} $ with $ p_n \rightarrow f $, then $ f(0) = 0 $. Since $ p_n(x) $ are odd polynomials, then $ p_n(0) = 0. $. Then $ f(0) = \lim_{n \rightarrow \infity} p_n(0) = 0 $
