f(f 1 (B)) B f(f 1 (B)) = B B f(s) f 1 (f(a)) A f 1 (f(a)) = A f : S T 若敘述為真則證明之, 反之則必須給反例 (Q, ) y > 1 y 1/n y t > 1 n > (y 1)/(t 1) y 1/n < t

Transcription

1 S T A S B T f : S T f(f 1 (B)) B f(f 1 (B)) = B B f(s) f 1 (f(a)) A f 1 (f(a)) = A f : S T f : S T S T f y T f 1 ({y) f(d 1 D 2 ) = f(d 1 ) f(d 2 ) D 1 D 2 S F x 0 x F x = 0 x = 0 x y = x y x, y F x + y x + y x, y F x y x y x, y F 若敘述為真則證明之, 反之則必須給反例 (Q, <) (Q, ) y > 1 y 1/n y y n 1 > n(y 1) n N y 1 > n(y 1/n 1) t > 1 n > (y 1)/(t 1) y 1/n < t

2 y 1/n = 1 n x n x n+1 x n 1 n x n F F x 0 ε > 0 x ε x = 0 S = (0, 1) ε > 0 x S x < ε A R A A { x R x A A = ( A). A, B R A + B = {x + y x A, y B (A + B) = A + B (A + B) = A + B (A B) { A, B (A B) = { A, B (A B) { A, B (A B) = { A, B S R S = { x R x S. b > 1 r, s Q b r+s = b r b s r, s Q b r s = (b r ) s

3 x R B(x) = { b t R t Q, t x b r = B(r) r Q b x = B(x) x R y > 0 u, v R b u < y b v > y b u+1/n < y b v 1/n > y n y > 0 A w b w < y x = A b x = y x 1, x 2 b x 1 = b x 2 x 1 = x 2 x b x = y b b y {x n n=1 R k N S k = {x k, x k+1,, x k+n, x n S k x n S k n k n k y k = x n z k = x n {y k k=1 n k n k {z k k=1 {x n n=1 k y k {x n n=1 k z k {x n n=1 k y k = k 1 y k {x n n=1 z k = z k k x n = ( 1) n( 1 1 n) k n k k 1 x n k n k x n A R { b B A <. b B { x A x 0 {a n n=1 {x n n=1 R S k = k n=1 a n {S k k=1 x n x n+1 < a n n N {x n n=1 {S k k=1 x y f : R R f(x) f(y) 2 x 1 R x k+1 = f(x k ) k N {x n n=1

4 {x n n=1 {y n n=1 R { x n y n n=1 x n x n {x n n=1 x {x n n=1 x n x n {x n n=1 x {x n n=1 R x n+1 ϵ n x n n N {x n n=1 k n=1 ϵ n n=1 ϵ n k π : N N π {x n n=1 { x π(n) n=1 {x n n=1 {y n n=1 R ( x n + y n x n )( (x n + y n ) y n ) (x n + y n ) x ny n ( x n y n ( x n + y n x n + x n )( x n )( y n ) y n ; y n ). x n+1 x n n xn n x n+1 xn. x n n x n x n+1 x n { m m = 0, 1, 2,

5 { (1 + 1 mπ ) m = 1, 2, m 6 α S = { x [0, 1] x = kα 1 k N [0, 1] y [0, 1] ε > 0 x S (y ε, y + ε) α = 1 2π T = { x [0, 2π] x = k 2π k N [0, 2π] S [0, 1] S k = { x [0, 1] x = lα 1 1 l k + 1 S k 1 k {x n n=1 R x n x n {x n n=1 M x n M {x n n=1 m x n m x n = {x n n=1 x n = {x n n=1 x {x n n=1 a = x n a b = x n b {x n n=1 x R x n x x n p R n ( n ) 1 x i p p 1 p <, x p i=1 { x 1,, x n p =, x n = x n = x R x = (x 1,, x n )

6 M n m : M R A = x R m x 0 Ax 2 x 2, 2 2 ( k ) 1/2 x 2 = x 2 i x = (x 1,, x k ) R k. i=1 A = Ax 2 = { M R Ax 2 M x 2 x R m x R m x 2 =1 Ax 2 A x 2 x R m M R U R U = k I(a k, b k ), I (a k, b k ) (a l, b l ) = k l (M, d) A M A A {U i i I A U i i I A U i i I { (a k, b k ) N (a k, b k ) [a, b] k=1 k=1 [a, b] R N > 0 A B (M, d) A)) = (A) (A B) = (A) (B) (A B) (A) (B) (A B) (A) (B)

7 A (M, d) (M, d) A M (A ) = A A B (M, d) A)) = (A) (A B) = (A) (B) (A B) (A) (B) (A B) (A) (B) (M, d) A M A = ( A (M\A) ) ( (A)\A ). A B (M, d) A = (M\A) ( A) (A) ( A) A (A B) A B (A B) A B (A) (B) = (A B) = A B ( ( A)) = ( A) (M, d) A M A = A\ A (A) = M\ (M\A) ( (A)) = (A) ( (A)) = A

8 ( (A)) = A A A M\A A A = (A)\ A A (M, d) A M A = A E 0 = [0, 1] ( 1 3, 2 ) E1 3 [ 1] [ 2 0,, 3 3, 1]. E 2 [ 1 [ 2 0,, 9] 9, 3 [ 6, 9] 9, 7 ] [ 8, 9 9, 1]. E k E 1 E 2 E 2 E n 2 n 3 n C = n=1 E n C C (C) (M, d) A S (A) A S Q R A S S T A T A S B S B (A B) x x x {x k k=1 d(x k, x) < ε ε > 0 N > 0 k > N

9 x {x k k=1 x x k x k {x k k=1 x x k x k {x k k=1 x (M, d) N M (N, d) N (M, d) N M (N, d) (M, d) {p n n=1 {q n n=1 M {p n n=1 {q n n=1 d(p n, q n ) = 0. {p n n=1 {p n n=1 {p n n=1 {q n n=1 {q n n=1 {p n n=1 {p n n=1 {q n n=1 {q n n=1 {r n n=1 {p n n=1 {r n n=1 {p n n=1 {q n n=1 { d(p n, q n ) n=1 R M P, Q M d (P, Q) = d(p n, q n ), {p n n=1 P {q n n=1 Q {p n n=1 P {q n n=1 Q d(p n, q n ) = d(p n, q n) φ : M M x M {x n n=1 x n x n N M {x n n=1 φ(x) φ(x) M φ(x) {x n n=1 d ( φ(x), φ(y) ) = d(x, y) x, y M. φ(m) M (M, d ) (M, d ) (M, d)

10 a n a n = n an n an a n n=1 a n+1 a n n + 1 n, 2 n n n. 3 n n n = n n + 1 = 1 a n+1 a n (M, d) A M d : R 2 R 2 R { x 1 y 1 x 2 = y 2, d(x, y) = x 1 y 1 + x 2 y x 2 y 2. x = (x 1, x 2 ) y = (y 1, y 2 ) d R 2 (R 2, d) D(x, r) r < 1 r = 1 r > 1 {c [a, b] (R 2, d) {c [a, b] (R 2, d) (M, d) A M (A) (M, d) K M K {F α α I K F α α I K α J F α J I J <. {x k k=1 x k k A {x 1, x 2,, {x x A A A (M, d)

11 M M X {x k k=1 X = { {x k k=1 xk R k N x k <. k 1 R x k < k 1 : X R {xk k=1 = x k. k 1 X (X, ) l (X, ) A, B, C, D X A = { {x k k=1 xk 1 k k N, B = { {x k k=1 xk 0 k, C = { {x k k=1 {xk k=1, D = { {x k k=1 x k = 1. k 1 A, B, C, D A, B R n d(a, B) = { x y 2 x A, y B A B A = {x d(a, B) d(x, B) d(a, B) = { d(x, B) x A d(x 1, B) d(x 2, B) x 1 x 2 2 x 1, x 2 R n B ε = { x R n d(x, B) < ε B ε B ε B ε = (B) A x A d(a, B) = d(x, B) A B x A y B d(a, B) = d(x, y) A B ε>0

12 M = { (x, y) R 2 x 2 + y A M A {x k k=1 R (R, ) x A k = {x k, x k+1, {x = A k k=1 {K j j=1 K j K j+1 (K j ) 0 j (K j ) = { d(x, y) x, y K j. K j (M, d) A M A F 1 F 2 A F 1 F 2 = A F 1 A F 2 A F 1 F 2 A (M, d) A B A B (M, d) A M A A A C x, y C x y x U y V U V C F k F k+1 F k F k k N F k k=1 F k n N R n (R n, 2 ) f : R 2 R x 0 y 0 j=1 f(x, y) f(x, y) y 0 x 0

13 f : R 2 R f f : R 2 R f : R 2 R (x, y) x U R A = { (x, y) R 2 x U f : R R U R f(u) f : A R m A R n B A f( (B) A) (f(b)). R n f : R n R f(x) = x f (R n, 2 ) f(x) f(y) C x y 2 C > 0 f : R n R m B R n f(b) f : R R K R f 1 (K) f : R R C R f 1 (C) K R n f : K R m f 1 : f(k) K K f : (0, ) R f(x) = x f : (0, 1) R f(x) = x 1 x f : (0, ) R f(x) = x

14 (M, d) (N, ρ) A M f : A N f A {x k k=1 {y k k=1 d(x k, y k ) 0 k ρ ( f(x k ), f(y k ) ) 0 k f A {x k k=1 { f(xk ) k=1 f : R R p > 0 f(x + p) = f(x) x R f f R a, b R f : (a, b) R f (a, b) f(x) f(x) x a + x b (a, b) f : [a, b] R α M > 0 α (0, 1] f(x 1 ) f(x 2 ) M x 1 x 2 α x 1, x 2 [a, b]. f [a, b] f : [0, ) R f(x) = x 1 2 (M, d) A M f, g : A R A f g fg A f g f : (a, b) R x 0 (a, b) m R f (x 0 ) ε > 0, δ > 0 f(x) f(x0 ) f (x 0 )(x x 0 ) ε x x0 x x 0 < δ. f, g : R R f 0 d dx f(x)g(x) α β β > 0 f : [ 1, 1] R { x α (x β ) x 0, f(x) = 0 x = 0.

15 f α > 0 f (0) α > 1 f α 1 + β f α > 1 + β f (0) α > 2 + β f α 2 + 2β f α > 2 + 2β f, g : [a, b] R g f 0 f fg x 0 (a, b) b a f(x)g(x)dx = g(x 0 ) b a f(x)dx. f : [a, b] R f b a f (x)dx = f(b) f(a) f : [a, b] R m f(x) M x [a, b] φ : [m, M] R φ f [a, b] A R f : A R f A I ε > 0 δ > 0 P = {x 0, x 1,, x n A δ {ξ 1,, ξ n ξ k [x k 1, x k ] k n f(ξ k )(x k+1 x k ) I < ε. k=1 n f(ξ k )(x k+1 x k ) f k=1 A f A