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; " 3$%5 " ! ; ! f (x) = f + (x) − f − (x), f + f − 3885 2 ! 3$%5 " 2 ! ! 3$%5 " & R f x 0 ∈ R * lim h →0+ f (x 0 + h) ( lim h →0− f (x 0 + h)) + f x 0 $& f (x 0 +0) (f(x 0 −0)) * f " x 0 3! &5 + f (x 0 + 0) = f(x 0 ) (f(x 0 ) = f(x 0 − 0)) f x 0 $ & " * f x 0 ! & + f (x 0 + 0) = f(x 0 ) = f(x 0 − 0) f x 0 * f (x 0 − 0) = f(x 0 + 0) f x 0 x 0 f Δ f (x) = f(x + 0) − f(x − 0) f x ∈ (a, b) f " ! Δ f (a) = f(a + 0) − f(a), Δ f (b) = f (b) − f(b − 0) * f x 0 ! & + ! f " < lim h →0+ f (x 0 + h) − f(x 0 ) h = Λ r , lim h →0+ f (x 0 − h) − f(x 0 ) h = Λ l , lim h →0+ f (x 0 + h) − f(x 0 ) h = λ r , lim h →0+ f (x 0 − h) − f(x 0 ) h = λ l , f x 0 [ . f (x) = [x] + 2 · sign (x + 1) + 3 · χ (−1,1] (x), [−2, 1] [−2, 1] , ! & , ! " & ! "# ' ! " 2 (; y 1 (x) = [x] ! " # y 2 (x) = sign (x+1) x = −1 ' ! & - lim h →0+ sign (0 − h) = −1 = sign 0 = 0 = lim h →0+ sign(0 + h) = 1. (9.4) y 2 (x) = sign (x + 1) y 2 (−1 + 0) − y 2 (−1 − 0) = 2 3$b5 ( x = −1 ! & 2( y 3 (x) = χ (−1,1] (x) [−2, 1] x = −1 ' x = −1 ! & # 1 + y 1 , y 2 y 3 & ! f (x) = [x] + 2 · sign (x + 1) + 3 · χ (−1,1] (x) [−2, 1] , −1 0 ' f (−1) = −1, f(0) = 5 lim h →0+ f (−1−h) = −2+2·(−1) = −4 = lim h →0+ f (−1+h) = −1+2·1+3·1 = 4, lim h →0+ f (0 − h) = −1 + 2 · 1 + 3 · 1 = 4 = lim h →0+ f (0 + h) = 0 + 2 · 1 + 3 · 1 = 5. ' −1 0 Δ f (−1) = 8 Δ f (0) = 1 ' x = −1 ! & x = 0 . f : [a, b] → R c 1 , c 2 , . . . , c n , n i =1 Δ f (c i ) ≤ f(b) − f(a) (9.5) f : [a, b] → R c 1 < c 2 < . . . < c n + < n i =1 Δ f (c i ) - n i =1 (f(c i + 0) − f(c i − 0)) = f(c 1 + 0) − f(c 1 − 0) + · · ·+ f(c n + 0) − f(c n − 0). ' ! - n i =1 Δ f (c i ) = f(c n + 0) − f(c 1 − 0) − n i =2 (f(c i − 0) − f(c i −1 + 0)). (9.6) = f c i > c i −1 ! f (c i − 0) − f(c i −1 + 0) ≥ 0 (9.7) 3$75 3$85 n i =1 Δ f (c i ) ≤ f(c n + 0) − f(c 1 − 0) ≤ f(b) − f(a) (9.8) ' 3$a5 . f : [a, b] → R n ∈ N , D n = x ∈ [a, b] : Δ f (x) ≥ 1 n ! & 1 n ∈ N ! D n & ! m ∈ N ! D n c 1 , c 2 , . . . , c m ∈ D n 3$a5 D n " - f (b) − f(a) ≥ m i =1 Δ f (c i ) ≥ m · 1 n . ' m ≤ n(f(b) − f(a)) ! ' m \ D n ! & . f : [a, b] → R c 1 , c 2 , . . . , c n , . . . , ∞ n =1 Δ f (c n ) ! ∞ n =1 Δ f (c n ) ≤ f(b) − f(a) = f : [a, b] → R [a, b] " ! c 1 , c 2 , . . . , c n ! 3 + " ! 5 n i =1 Δ f (c i ) ≤ f(b) − f(a) %" 33$W5 5 ' ∞ n =1 Δ f (c n ) " " S n ! \ ∞ n =1 Δ f (c n ) ! 3$W5 n → ∞ ∞ n =1 Δ f (c n ) ≤ f(b) − f(a) .$ f : [a, b] → R c 1 , c 2 , . . . f d (x) = c i ≤x Δ f (c i ), x ∈ [a, b] (9.9) f f d : [a, b] → R = f : [a, b] → R c 1 , c 2 , . . . + ! ( c 1 < c 2 < · · · < c n < · · · 3$$5 f d (x) = c i ≤x Δ f (c i ), x ∈ [a, b] [a, b] " x 1 < x 2 ∈ [a, b] , f c ∈ [a, b] ! Δ f (c) ≥ 0 " f d (x 1 ) = c i ≤x 1 Δ f (c i ) ≤ c i ≤x 1 Δ f (c i ) + x 1 j ≤x 2 Δ f (c j ) = f d (x 2 ) ' f : [a, b] → R 2 ( f f d c 1 , c 2 , . . . f f d " c i ( Δ f (c i ) = Δ f d (c i ). (c i , c i +1 ) f d .% f (x) = x + 2[x], x ∈ [0, 2] [0, 2] " & ! "# ( x − [x] " g (x) = x ! \ f " , x 1 = 1 x 2 = 2 Δ f (x 1 ) = Δ f (x 2 ) = 2 ' f d (x) = 2[x] f c (x) = x +4 # "4-4- ,-0) ( / 3$%5 ! ; (; ! ; " . 7, + [a, b] f ! [a, b] a = x 0 < x 1 < · · · < x n −1 < x n = b n x i (i = 1, 2, . . . , n) n k =1 Download 1.57 Mb. Do'stlaringiz bilan baham: 
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