Define a new norm r on S as r(x) = sqrt(|x|2 + |f(x)|2). This is basically the euclidean formula for distance in two dimensions, so it satisfies the properties of a norm.
Start with a cauchy sequence under the norm r, and select ε and n so that terms beyond xn differ by no more than ε. Since distance under |S| or |T| is never larger than distance under r(), the terms beyond xn differ by no more than ε under |S|, and their images differ by no more than ε under |T|. Both xn and f(xn) are cauchy, and converge to x and y. By assumption y = f(x), so x is the limit of the sequence under the metric r(). In other words, S is still a complete banach space.
Apply the identity map on S from |S| to r(S). The former metric is always bounded by the latter, so by the previous theorem, the map is a homeomorphism, continuous in both directions, and there is a reverse bound b satisfying r(x) ≤ b×|x|. Yet the metric in T is bounded by the metric in r, so distance in T is bounded by some constant times distance in S, and f is continuous.