Determining if this subset is a subring

Hi there. Firstly, sorry if the title wasn't overly descriptive.

I'm working on an assignment question dealing with group/ring theory that is:

Let be the ring of all real-valued functions defined on the closed interval [0, 1]. Decide if the folowing subset of R is a subring of R:

has an infinite number of solutions in [0, 1]

I know since it is already a subset of R that I need to check if:

for if

and

So basically whether

and

will also have an infinite number of solutions.

Another question was similar but the subset consisted of the functions of R that had only a finite number of solutions in [0,1]. I was able to determine by contradiction using an example that it indeed was not, so my intuition here is that this subset is a subring. I'm just not sure how to start it.

Thanks for any help,

--

Dave