On a Class of Pathological Functions

We study the problem of approximating a function from a separable metric space D to a separable metric space Z by functions g from D to Z which have the property there is no set E of power 2** such that g is a homeomorphism of E onto g(E). Theorem 1 asserts that any one to one function may be approximated by such a function. Numerous continuous functions may be approximated by such functions which, moreover, are periodic of preassigned period n (Theorem 4). All functions mentioned are from a separable metric space into a separable metric space. Definition. A one to one function f of D into Z is a dishomeomorphism if D contains no subset E, of power 2Ko, such that/ is a homeomorphism of E onto f(E). The proof of the existence of a dishomeomorphism was given for the first time in [2]. By R is meant the set of real numbers, ordered in the natural manner. By a linear set is meant a subset of R. Definition. A one to one function / of a linear set E into R is a dissimilarity transformation (on E) if E contains no subset D, of power 2No, such that/ is a similarity transformation of D onto f(D). From the definitions there readily follows