ABSTRACT

For holomorphic functions of several variables the situation is much more complicated, but some knowledge of the local properties is essential as a background to any study of holomorphic functions of several variables. The construction of germs of functions can be carried out by considering continuous functions rather than arbitrary complex-valued functions. The resulting ring is then called the ring of germs of continuous complex-valued functions. The ring operations both arise from pointwise addition and multiplication of the functions represented, and that is enough to conclude the proof. The proof is actually just an extension of the proof of the implicit function theorem.