Solve parts (c), (d) and (e) Suppose fR → R is a function such that for all x, y € R, f(x + y) = f(x) + f(y) andf(xy) = f(x)f(y). We proved in Tutorial 4B that then we also have f(0) = 0, f(1) = 1,f(-1) = -1, and the implication a < b ⇒ f(a) < f(b). You may freely use these facts insolving this question.(a) Show that for each n = Z, we have f(n) = n.(b) Show that for each z EQ, we have f(x) = x.(c) Let A CR, and write f(A) for the set {r Ry & A: f(y) = x}. Show: if A isbounded above, then f(A) is bounded above.(d) Let A CR be bounded above, with supremum L. Show that f(L) is the supremum off(A).(e) Now use the statements proved in the previous parts to prove: for all a R, we havef(x) = x.

You asked for c),d) and e) so, here it is

(c) Let A CR, and write f(A) for the set { € Ry € A : f(y) = x}. Show: if A is bounded above, then f(A) is bounded above. (d) Let A CR be bounded above, with supremum L. Show that f(L) is the supremum of f(A). (e) Now use the statements proved in the previous parts to prove: for all a R, we have f(x) = x.