On the Continuity of the Real Roots of an Algebraic Equation

It is well known that the root of an algebraic equation is a continuous multiple-valued function of its coefficients [5, p. 3]. However, it is not necessarily true that a root can be given by a continuous single-valued function. A complete solution of this problem has long been known in the case where the coefficients are themselves polynomials in a complex variable [3, chap. V]. For most purposes the concept of the Riemann surface enables one to bypass the problem. However, in the study of the ideal structure of rings of continuous functions, the general problem must be met directly. This paper is confined to an investigation of the continuity of the real roots of an algebraic equation; the results obtained are used to establish a theorem stated, but not correctly proved, by Hewitt [2, Theorem 42] on rings of real-valued continuous functions