Are the following subrings of the ring of all functions from the closed interval $\displaystyle [0,1] $ to $\displaystyle \mathbb{R}$:

1. the set of all functions which have only a finite number of zeros, together with the zero function

2. the set of all functions $\displaystyle f$ such that $\displaystyle lim_{x -> 1^{-}} f(x) = 0$