- Problem 4 A Let F G Be Continuous Mappings From R Into R Further Let D Be A Dense Subset Of R Show That F D Is De 1 (36.48 KiB) Viewed 17 times
Problem 4 (a) Let f, g be continuous mappings from R into R. Further, let D be a dense subset of R. Show that f(D) is dense in f(R). Furthermore, show that if f(x) = g(x) for all x E D, then f(z) = g(z) for all z E R (This shows that a continuous function f:R - R is uniquely determined by its values on D.) = + = (b) Suppose f:R → R is a continuous function such that f(x + y) = f(x) + f(y) for all x, y € R. Show that f is of the form f(x) = c.x for some c ER.