Summary: The Weierstrass theory is strengthened as “the continuous functions \(f(x)\) on a closed interval can be approximated by polynomials with rational coffcients” first. Then the relation between the continuous function \(f(x)\) on \([a,b]\) and polynomial series is established, and the \(1-1\) mapping from the set of continuous functions on \([a,b]\) to the set of rational number sequences is given by the polynomial series. Furthermore, the famous result on “the set of continuous functions on \([a,b]\) has the potency of the continuum” is obtained.