5/19/2018 Articulo de La Integral de Cauchy a La Integral de Riemann Omar Garcia

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f

[a, b]

lmx0

n

i=1

f(xi)(xixi1)

{x0, x1, x2,...,xn} [a, b] x

[a, b]

f : [a, b]R

f

[a, b]

x [a, b]

5/19/2018 Articulo de La Integral de Cauchy a La Integral de Riemann Omar Garcia

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f(x) = lmtx

f(t) f(x+) = lmtx+

f(t)

x = a

f(a+)

x = b

f(b) G[a, b] [a, b]

f : [a, b] R p [a, b]

f(p) = sup {f(x)|x [a, p)} f(p+) = nf{f(x)|x (p, b]}

f(p) f(p) f(p+) q, r [a, b] q < r f(q+) f(r)

f(p) = nf{f(x)|x [a, p)} f(p+) = sup {f(x)|x (p, b]}

f(p+) f(p) f(p) q, r [a, b] q < r f(r) f(q+).

x= a f(a+) x= b f(b)

E[a, b]

C[a, b]

2,1

f G[a, b] >0 E[a, b]

|f(x)(x)|< , x [a, b]

f 0 [a, b] 0

f G[a, b] {n}nN E[a, b] f [a, b]

2,3

f G[a, b] f

f G[a, b] {n(x)}nN E[a, b] n(x) f [a, b]

ba

n(x) dx

nN

{n(x)}nN E[a, b] n(x) f [a, b]

>0

n0 N n > n0 |n(x)n(x)|< , x [a, b]

lmn

ba

n(x) dx = lmn

ba

n(x) dx

5/19/2018 Articulo de La Integral de Cauchy a La Integral de Riemann Omar Garcia

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f G[a, b] f [a, b]

lmn ba n(x) dx

{n(x)}nN E[a, b] n(x) f [a, b] f C

ba

f(x) dxC

C

f [a, b]

U(f) ={ (x) E[a, b]|f(x) (x), x [a, b]}

L(f) ={ (x) E[a, b]|(x) f(x), x [a, b]}

f U(f) L(f) (x) L(f) (x) U(f)

(x) f(x)

(x), x [a, b]

U=

ba

(x) dx| (x) U(f)

; L=

ba

(x) dx|(x) L(f)

nf (x)U(f)

U sup (x)L(f)

L

f [a, b]

b

a

f(x) dxR

b

a

f(x) dxR

f R b

a

f(x) dxR=

ba

f(x) dxR

f [a, b]

ba

f(x) dxR

5/19/2018 Articulo de La Integral de Cauchy a La Integral de Riemann Omar Garcia

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f : [a, b] R R >0 (x), (x) E[a, b]

(x) f(x) (x), x [a, b] ba

(x) dx ba

(x) dx <

f C f R

f g [a, b]

f G[a, b] g(x) =g(a) +

xa

f(t) dtC x [a, b]

x [a, b) g+(x) =f(x+) x (a, b] g(x) =f(x)

f G[a, b] g f g C[a, b] [a, b] g(x) g(x) =f(x)

f C g: [a, b] R

g(x) =g(a) +

xa

f(t) dtCparacadax [a, b]

f c [a, b] g c

g(c) =f(c)

c= a c= b g(c) g a g

f C

g: [a, b] R

g

(x) =f(x)

x [a, b] b

a

f(x) dxC =g(b)g(a)

5,2

5,3

f

C R

5/19/2018 Articulo de La Integral de Cauchy a La Integral de Riemann Omar Garcia

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