Suppose fR → R is a function such that for all x, y € R, f(x + y) = f(x) + f(y) and f(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 in solving 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 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 z R, we have f(x) = x.
Suppose fR → R is a function such that for all x, y € R, f(x + y) = f(x) + f(y) and f(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 in solving 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 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 z R, we have f(x) = x.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose fR → R is a function such that for all x, y € R, f(x + y) = f(x) + f(y) and
f(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 in
solving 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 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.
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