On some properties of approximations

Let f:X→Y be continuous where X is a topological space and Y a metric space. Given a set E⊂Y, we ask whether f admits arbitrarily close continuous approximations whose values omit E (see Definition 2). It is shown that if X is paracompact, , then each continuous mapping X→Rn, n>k, has an arbitrarily close approximation avoiding the product of n given boundary subsets of R.Also, we discuss a related topic consisting in finding conditions under which the approximating mappings do not take values in certain balls. In this connection, we investigate relations between the accuracy of approximations and the radii of omitted balls.