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**topsquark** Okay, I can prove that the example I mentioned above is a ring. This is one of a six part problem (not related to each other) that gives the ring of functions $\displaystyle f: [0,~1] \to \mathbb{R}$, the purpose of which seems to be listing different kinds of subrings. The reason I was thinking that the one I posted may not be a subring is because the restriction $\displaystyle \lim_{x \to 1^-} f(x) = 0$ seems so trivial that I was thinking there was some kind of "trick" to it. Thinking too hard about it, I guess.

However there is still a potential problem to my mind. All of these problems deal with subrings that map [0, 1] to $\displaystyle \mathbb{R}$, where we have + and $\displaystyle \cdot$ defined as usual for real numbers. I can't seem to find a set of functions mapping to $\displaystyle \mathbb{R}$ that wouldn't be a subring. And that makes no sense to me.

-Dan