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 $\displaystyle R=\{f : [0, 1] \to \mathbb{R}\}$ 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:

$\displaystyle S = \{f \in R | f(x) = 0$ has an infinite number of solutions in [0, 1]$\displaystyle \}$

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

for $\displaystyle g, h \in S$ if

$\displaystyle g - h \in S$ and $\displaystyle gh \in S$

So basically whether

$\displaystyle g - h = 0$ and $\displaystyle gh = 0$

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,

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Dave