4.2.9: (a) Because there is no restriction that the function be given by a single formula, or even that it be continuous, two extensions are given by (i) f(x)=x

^{2}for x in R, and (ii) f(x)= x^{2}for x in N and f(x) = 0 for x in R \ N. Since N and R \ N have empty intersection and union R, the second formula defines a function f : R ---> R.Similar "piecewise-defined" functions can be used for the deisred extensions in parts (b) and (c).