TITLE: Embedding of continuous functions into differentiable functions
QUESTION [7 upvotes]: This question refers to a solution printed in the current (December 2013, 120(10)) issue of The American Mathematical Monthly, p. 944.  There, the authors intend to show that any ring homomorphism from the set $C$ of continuous functions R $\to$ R to the set of differentiable functions R $\to$ R cannot be an injection.  (Note we are talking of only an algebraïc homomorphism.  That is, we have no topology on the function spaces.)
I follow the solution through "the image of [such a homomorphism] $\phi$ consists only of constant functions."  What I don't get is the following (and terminal) sentence, "In particular, $\phi$ is not injective."  This does not follow from cardinality considerations, as $C$ has the same cardinality as R.  Could someone please elucidate?

REPLY [3 votes]: Incidentally, I solved that problem (though the "Also solved by" list didn't appear, damn it!!). My final argument was: If $\phi$ were injective, then $C$ would be isomorphic to a subring of $\mathbb R$, which will be an integral domain; but clearly $C$ is not an integral domain.