Let X be a Tychonoff space and

the set of all the subrings of C(X) that contain

. For any A(X) in

suppose

is the largest subspace of

containing X to which each function in A(X) can be extended continuously. Let us write A(X) ~ B(X) if and only if

, thereby defining an equivalence relation on

. We have shown that an A(X) in

is isomorphic to C(Y ) for some space Y if and only if A(X) is the largest member of its equivalence class if and only if there exists a subspace T of

with the property that A(X)={

is real for each

in T},

being the unique continuous extension of

in C(X) from

to

, the one point compactification of

. As a consequence it follows that if X is a realcompact space in which every

-embedded subset is closed, then C(X) is never isomorphic to any A(X) in

without being equal to it.