:: The Limit of a Real Function at a Point
:: by Jaros{\l}aw Kotowicz
::
:: Received September 5, 1990
:: Copyright (c) 1990 Association of Mizar Users
Lm1:
for g, r, r1 being real number st 0 < g & r <= r1 holds
( r - g < r1 & r < r1 + g )
Lm2:
for seq being Real_Sequence
for f1, f2 being PartFunc of REAL , REAL
for X being set st rng seq c= (dom (f1 (#) f2)) \ X holds
( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X )
Lm3:
for r being Real
for n being Element of NAT holds
( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) )
Lm4:
for seq being Real_Sequence
for f1, f2 being PartFunc of REAL , REAL
for X being set st rng seq c= (dom (f1 + f2)) \ X holds
( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X )
theorem Th1: :: LIMFUNC3:1
theorem Th2: :: LIMFUNC3:2
theorem Th3: :: LIMFUNC3:3
theorem Th4: :: LIMFUNC3:4
theorem Th5: :: LIMFUNC3:5
theorem Th6: :: LIMFUNC3:6
theorem Th7: :: LIMFUNC3:7
theorem Th8: :: LIMFUNC3:8
for
x0 being
Real for
f being
PartFunc of
REAL ,
REAL holds
( ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) iff ( ( for
r being
Real st
r < x0 holds
ex
g being
Real st
(
r < g &
g < x0 &
g in dom f ) ) & ( for
r being
Real st
x0 < r holds
ex
g being
Real st
(
g < r &
x0 < g &
g in dom f ) ) ) )
:: deftheorem Def1 defines is_convergent_in LIMFUNC3:def 1 :
:: deftheorem Def2 defines is_divergent_to+infty_in LIMFUNC3:def 2 :
:: deftheorem Def3 defines is_divergent_to-infty_in LIMFUNC3:def 3 :
theorem :: LIMFUNC3:9
canceled;
theorem :: LIMFUNC3:10
canceled;
theorem :: LIMFUNC3:11
canceled;
theorem :: LIMFUNC3:12
theorem :: LIMFUNC3:13
theorem :: LIMFUNC3:14
theorem Th15: :: LIMFUNC3:15
theorem Th16: :: LIMFUNC3:16
theorem :: LIMFUNC3:17
theorem :: LIMFUNC3:18
theorem :: LIMFUNC3:19
theorem :: LIMFUNC3:20
theorem :: LIMFUNC3:21
theorem :: LIMFUNC3:22
theorem Th23: :: LIMFUNC3:23
theorem :: LIMFUNC3:24
theorem Th25: :: LIMFUNC3:25
theorem :: LIMFUNC3:26
theorem Th27: :: LIMFUNC3:27
for
x0 being
Real for
f1,
f being
PartFunc of
REAL ,
REAL st
f1 is_divergent_to+infty_in x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) & ex
r being
Real st
(
0 < r &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f1 . g <= f . g ) ) holds
f is_divergent_to+infty_in x0
theorem Th28: :: LIMFUNC3:28
for
x0 being
Real for
f1,
f being
PartFunc of
REAL ,
REAL st
f1 is_divergent_to-infty_in x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) & ex
r being
Real st
(
0 < r &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= f1 . g ) ) holds
f is_divergent_to-infty_in x0
theorem :: LIMFUNC3:29
theorem :: LIMFUNC3:30
:: deftheorem Def4 defines lim LIMFUNC3:def 4 :
theorem :: LIMFUNC3:31
canceled;
theorem :: LIMFUNC3:32
theorem Th33: :: LIMFUNC3:33
theorem :: LIMFUNC3:34
theorem Th35: :: LIMFUNC3:35
theorem Th36: :: LIMFUNC3:36
theorem Th37: :: LIMFUNC3:37
theorem :: LIMFUNC3:38
theorem :: LIMFUNC3:39
theorem :: LIMFUNC3:40
theorem Th41: :: LIMFUNC3:41
theorem Th42: :: LIMFUNC3:42
theorem :: LIMFUNC3:43
theorem :: LIMFUNC3:44
theorem Th45: :: LIMFUNC3:45
for
x0 being
Real for
f1,
f2,
f being
PartFunc of
REAL ,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in x0 &
lim f1,
x0 = lim f2,
x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) & ex
r being
Real st
(
0 < r & ( for
g being
Real st
g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
(
f1 . g <= f . g &
f . g <= f2 . g ) ) & ( (
(dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) or (
(dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) ) ) holds
(
f is_convergent_in x0 &
lim f,
x0 = lim f1,
x0 )
theorem :: LIMFUNC3:46
for
x0 being
Real for
f1,
f2,
f being
PartFunc of
REAL ,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in x0 &
lim f1,
x0 = lim f2,
x0 & ex
r being
Real st
(
0 < r &
].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for
g being
Real st
g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds
(
f1 . g <= f . g &
f . g <= f2 . g ) ) ) holds
(
f is_convergent_in x0 &
lim f,
x0 = lim f1,
x0 )
theorem :: LIMFUNC3:47
for
x0 being
Real for
f1,
f2 being
PartFunc of
REAL ,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in x0 & ex
r being
Real st
(
0 < r & ( (
(dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f1 . g <= f2 . g ) ) or (
(dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f1 . g <= f2 . g ) ) ) ) holds
lim f1,
x0 <= lim f2,
x0
theorem :: LIMFUNC3:48
theorem :: LIMFUNC3:49
theorem :: LIMFUNC3:50
theorem :: LIMFUNC3:51
theorem :: LIMFUNC3:52