CONNECTED AND DISCONNECTED PLANE SETS AND THE FUNCTIONAL EQUATION f(x)+f(y)=f(x+y)

Cauchy discovered before 1821 that a function satisfying the equation ƒ(*)+ƒ (y) =f(* + y) is either continuous or totally discontinuous. After Hamel showed the existence of a discontinuous function satisfying the equation, many mathematicians have concerned themselves with problems arising from the study of such functions. However the following question seems to have gone unanswered : Since the plane image of such a function (the graph of y =f(x)) must either be connected or be totally disconnected, must the function be continuous if its image is connected? The answer is no. The utility of this answer is at once apparent. For if f(x) is totally discontinuous, its image obviously contains neither a continuum nor (in view of Darboux's work) a bounded connected subset even if the image itself is connected. As a matter of fact, if f(x) is discontinuous but its image is connected, then the image, its complement, or some simple modification thereof, serves to illustrate rather easily many of the strange and non-intuitive properties of connected sets now illustrated by numerous complicated examples scattered through the literature. Thus this class of sets is a useful tool in studying connectedness and disconnectedness. A few illustrations are given, particularly in connection with