Are the following subrings of the ring of all functions from the closed interval to :
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 such that
For the first one, think about what kind of functions have finite zeros or are constant zero. The polynomials form a ring, but are there other elements in this subring?
For the second, simply check the subring properties. If you add two functions with this limit property, does the result also have this limit? It is straightforward.