# Section 3.3: Problem 8 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Let $g$ and $h$ be representable functions, and assume that Show that $f$ is representable.
One way is to use the primitive recursion (catalog item 13). Let $d(n,0,b)=g(b)$ , and $d(n,a+1,b)=h((n)_{a},a,b)$ . Then, $f(0,b)=g(b)=d(\langle\rangle,0,b)$ , and $f(a+1,b)=h(f(a,b),a,b)=d(\overline{f}(a+1,b),a+1,b)$ , therefore, $f$ is the function defined by $d$ by primitive recursion. Therefore, it is sufficient to show that $d$ is representable. $d(n,0,b)=g(I_{3}^{3}(n,0,b))$ is representable, and for $a>0$ , $d(n,a,b)=h((I_{1}^{3}(n,a,b))_{I_{2}^{3}(n,a,b)-1},I_{2}^{3}(n,a,b),I_{3}^{3}(n,a,b))$ is representable. Hence, $g$ is representable (a more general case of “gluing” representable functions together is presented in Exercise 10).