:: More on the Continuity of Real Functions
:: by Keiko Narita , Artur Kornilowicz and Yasunari Shidama
::
:: Received February 22, 2011
:: Copyright (c) 2011-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, SEQ_1, PARTFUN1, RELAT_1, TARSKI, SEQ_2,
ORDINAL2, FUNCT_2, FUNCT_1, XBOOLE_0, XXREAL_0, NAT_1, ARYTM_3, CARD_1,
COMPLEX1, ARYTM_1, REAL_1, RCOMP_1, VALUED_1, ZFMISC_1, XXREAL_2,
FCONT_1, NORMSP_1, STRUCT_0, PRE_TOPC, FINSEQ_1, CARD_3, FINSET_1,
MEMBERED, BORSUK_1, REAL_NS1, XCMPLX_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1,
RELSET_1, PARTFUN1, FUNCT_2, FINSET_1, NUMBERS, XCMPLX_0, XXREAL_0,
XREAL_0, VALUED_1, REAL_1, MEMBERED, COMPLEX1, XXREAL_2, FINSEQ_1, SEQ_1,
SEQ_2, RCOMP_1, FCONT_1, STRUCT_0, PRE_TOPC, RLVECT_1, NORMSP_0,
NORMSP_1, EUCLID, NFCONT_1, REAL_NS1, PDIFF_1, INTEGR15, VALUED_2,
NFCONT_3, VFUNCT_1;
constructors REAL_1, COMPLEX1, SEQ_2, SEQ_4, RELSET_1, FCONT_1, NFCONT_1,
VFUNCT_1, BINOP_2, PDIFF_1, INTEGR15, NFCONT_3, VALUED_2, COMSEQ_2;
registrations XBOOLE_0, ORDINAL1, RELSET_1, NUMBERS, XXREAL_0, XREAL_0, NAT_1,
VALUED_0, FUNCT_2, XXREAL_2, FUNCT_1, STRUCT_0, EUCLID, FINSEQ_1,
REAL_NS1, NFCONT_3, VALUED_2;
requirements REAL, NUMERALS, SUBSET, BOOLE;
begin :: Basic Properties of the continuous functions of PartFunc of REAL,REAL n
reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;
definition let n,f,x0;
pred f is_continuous_in x0 means
:: NFCONT_4:def 1
ex g be PartFunc of REAL,REAL-NS n st f=g & g is_continuous_in x0;
end;
theorem :: NFCONT_4:1
h = f implies (f is_continuous_in x0 iff h is_continuous_in x0);
theorem :: NFCONT_4:2
x0 in X & f is_continuous_in x0 implies f|X is_continuous_in x0;
theorem :: NFCONT_4:3
f is_continuous_in x0 iff x0 in dom f &
for r st 0 PartFunc of Z,REAL means
:: NFCONT_4:def 2
dom it = dom f & for x be set st x in dom it holds it/.x = |. f/.x .|;
end;
definition
let n be Element of NAT;
let Z be non empty set;
let f be PartFunc of Z,REAL n;
func -f -> PartFunc of Z, REAL n means
:: NFCONT_4:def 3
dom it = dom f & for c be set st c in dom it holds it/.c = - (f/.c);
end;
theorem :: NFCONT_4:5
for f1,f2 be PartFunc of W,REAL-NS n,
g1,g2 be PartFunc of W,REAL n st f1=g1 & f2=g2 holds
f1+f2 = g1+g2;
theorem :: NFCONT_4:6
for f1 be PartFunc of W,REAL-NS n,
g1 be PartFunc of W,REAL n,
a be Real st f1=g1 holds
a(#)f1 = a(#)g1;
theorem :: NFCONT_4:7
for f1 be PartFunc of W,REAL n holds (-1)(#)f1 = -f1;
theorem :: NFCONT_4:8
for f1 be PartFunc of W,REAL-NS n,
g1 be PartFunc of W,REAL n st f1=g1 holds
-f1 = -g1;
theorem :: NFCONT_4:9
for f1 be PartFunc of W,REAL-NS n,
g1 be PartFunc of W,REAL n st f1=g1 holds
||. f1 .|| = |. g1 .|;
theorem :: NFCONT_4:10
for f1,f2 be PartFunc of W,REAL-NS n,
g1,g2 be PartFunc of W,REAL n st f1=g1 & f2=g2 holds
f1-f2 = g1-g2;
theorem :: NFCONT_4:11
f is_continuous_in x0 iff
x0 in dom f & for N1 be Subset of REAL n st ex r be Real st 0 < r &
{y where y is Element of REAL n: |.y-f/.x0.| < r} = N1
ex N being Neighbourhood of x0 st
for x1 st x1 in dom f & x1 in N holds f/.x1 in N1;
theorem :: NFCONT_4:12
f is_continuous_in x0 iff x0 in dom f &
for N1 be Subset of REAL n
st ex r be Real st 0 < r &
{y where y is Element of REAL n: |.y-f/.x0.| < r} = N1
ex N being Neighbourhood of x0 st f.:N c= N1;
theorem :: NFCONT_4:13
(ex N be Neighbourhood of x0 st dom f /\ N = {x0}) implies
f is_continuous_in x0;
theorem :: NFCONT_4:14
x0 in dom f1 /\ dom f2 & f1 is_continuous_in x0 & f2 is_continuous_in x0
implies f1+f2 is_continuous_in x0;
theorem :: NFCONT_4:15
x0 in dom f1 /\ dom f2 & f1 is_continuous_in x0 & f2 is_continuous_in x0
implies f1-f2 is_continuous_in x0;
theorem :: NFCONT_4:16
f is_continuous_in x0 implies r(#)f is_continuous_in x0;
theorem :: NFCONT_4:17
x0 in dom f & f is_continuous_in x0 implies
|. f .| is_continuous_in x0;
theorem :: NFCONT_4:18
x0 in dom f & f is_continuous_in x0 implies
-f is_continuous_in x0;
theorem :: NFCONT_4:19
for S be RealNormSpace,
z be Point of REAL-NS n,
f1 be PartFunc of REAL,REAL n,
f2 be PartFunc of REAL-NS n,S
st x0 in dom (f2*f1) & f1 is_continuous_in x0
& z=f1/.x0 & f2 is_continuous_in z
holds f2*f1 is_continuous_in x0;
theorem :: NFCONT_4:20
for S be RealNormSpace,
f1 be PartFunc of REAL,S,
f2 be PartFunc of S,REAL
st x0 in dom (f2*f1) & f1 is_continuous_in x0
& f2 is_continuous_in f1/.x0
holds f2*f1 is_continuous_in x0;
definition let n;
let f be PartFunc of REAL n,REAL;
let x0 be Element of REAL n;
pred f is_continuous_in x0 means
:: NFCONT_4:def 4
ex y0 be Point of REAL-NS n, g be PartFunc of REAL-NS n,REAL st
x0=y0 & f=g & g is_continuous_in y0;
end;
theorem :: NFCONT_4:21
for f be PartFunc of REAL n,REAL,
h be PartFunc of REAL-NS n,REAL,
x0 be Element of REAL n,
y0 be Point of REAL-NS n
st f=h & x0=y0 holds
f is_continuous_in x0 iff h is_continuous_in y0;
theorem :: NFCONT_4:22
for f1 be PartFunc of REAL,REAL n,
f2 be PartFunc of REAL n,REAL st x0 in dom (f2*f1) &
f1 is_continuous_in x0 & f2 is_continuous_in f1/.x0
holds f2*f1 is_continuous_in x0;
definition let n,f;
attr f is continuous means
:: NFCONT_4:def 5
for x0 st x0 in dom f holds f is_continuous_in x0;
end;
theorem :: NFCONT_4:23
for g be PartFunc of REAL,REAL-NS n,
f be PartFunc of REAL,REAL n st g=f holds
g is continuous iff f is continuous;
theorem :: NFCONT_4:24
X c= dom f implies
(f|X is continuous iff for x0,r st x0 in X & 0 continuous for PartFunc of REAL,REAL n;
end;
registration let n;
cluster continuous for PartFunc of REAL,REAL n;
end;
registration let n;
let f be continuous PartFunc of REAL,REAL n, X be set;
cluster f|X -> continuous for PartFunc of REAL,REAL n;
end;
theorem :: NFCONT_4:25
f|X is continuous & X1 c= X implies f|X1 is continuous;
registration let n;
cluster empty -> continuous for PartFunc of REAL,REAL n;
end;
registration let n,f;
let X be trivial set;
cluster f|X -> continuous for PartFunc of REAL,REAL n;
end;
registration let n;
let f1,f2 be continuous PartFunc of REAL,REAL n;
cluster f1+f2 -> continuous for PartFunc of REAL,REAL n;
end;
theorem :: NFCONT_4:26
X c= dom f1 /\ dom f2 & f1|X is continuous & f2|X is continuous
implies (f1+f2) |X is continuous & (f1-f2) |X is continuous;
theorem :: NFCONT_4:27
X c= dom f1 & X1 c= dom f2 & f1|X is continuous & f2|X1 is continuous
implies (f1+f2) | (X /\ X1) is continuous & (f1-f2) | (X /\ X1) is continuous
;
registration let n;
let f be continuous PartFunc of REAL,REAL n;
let r;
cluster r(#)f -> continuous for PartFunc of REAL,REAL n;
end;
theorem :: NFCONT_4:28
X c= dom f & f|X is continuous implies (r(#)f) | X is continuous;
theorem :: NFCONT_4:29
X c= dom f & f|X is continuous
implies |. f .| |X is continuous & (-f) | X is continuous;
theorem :: NFCONT_4:30
f is total & (for x1,x2 holds f/.(x1+x2) = f/.x1 + f/.x2) &
(ex x0 st f is_continuous_in x0) implies f|REAL is continuous;
theorem :: NFCONT_4:31
for Y be Subset of REAL-NS n st
dom f is compact & f| (dom f) is continuous & Y = rng f
holds Y is compact;
theorem :: NFCONT_4:32
for Y be Subset of REAL, Z be Subset of REAL-NS n st
Y c= dom f & Z = f.:Y & Y is compact & f|Y is continuous
holds Z is compact;
definition let n,f;
attr f is Lipschitzian means
:: NFCONT_4:def 6
ex g be PartFunc of REAL,REAL-NS n st g=f & g is Lipschitzian;
end;
theorem :: NFCONT_4:33
f is Lipschitzian iff
ex r be Real st 0 Lipschitzian for PartFunc of REAL,REAL n;
end;
registration let n;
cluster empty for PartFunc of REAL,REAL n;
end;
registration let n;
let f be Lipschitzian PartFunc of REAL,REAL n, X be set;
cluster f|X -> Lipschitzian for PartFunc of REAL,REAL n;
end;
theorem :: NFCONT_4:36
f|X is Lipschitzian & X1 c= X implies f|X1 is Lipschitzian;
registration let n;
let f1,f2 be Lipschitzian PartFunc of REAL,REAL n;
cluster f1+f2 -> Lipschitzian for PartFunc of REAL,REAL n;
cluster f1-f2 -> Lipschitzian for PartFunc of REAL,REAL n;
end;
theorem :: NFCONT_4:37
f1|X is Lipschitzian & f2|X1 is Lipschitzian implies
(f1+f2) | (X /\ X1) is Lipschitzian;
theorem :: NFCONT_4:38
f1|X is Lipschitzian & f2|X1 is Lipschitzian implies
(f1-f2) | (X /\ X1) is Lipschitzian;
registration let n;
let f be Lipschitzian PartFunc of REAL, REAL n;
let p;
cluster p(#)f -> Lipschitzian for PartFunc of REAL, REAL n;
end;
theorem :: NFCONT_4:39
f|X is Lipschitzian & X c= dom f implies (p(#)f) | X is Lipschitzian;
registration let n;
let f be Lipschitzian PartFunc of REAL, REAL n;
cluster |. f .| -> Lipschitzian for PartFunc of REAL,REAL;
end;
theorem :: NFCONT_4:40
f|X is Lipschitzian implies -(f|X) is Lipschitzian
& |. f .| |X is Lipschitzian & (-f) |X is Lipschitzian;
registration let n;
cluster constant -> Lipschitzian for PartFunc of REAL, REAL n;
end;
registration let n;
cluster Lipschitzian -> continuous for PartFunc of REAL, REAL n;
end;
theorem :: NFCONT_4:41
for r,p be Element of REAL n
st (for x0 st x0 in X holds f/.x0 = x0*r+p)
holds f|X is continuous;
theorem :: NFCONT_4:42
for x0 be Element of REAL n st 1 <= i & i <= n
holds proj(i,n) is_continuous_in x0;
theorem :: NFCONT_4:43
for n be non zero Element of NAT, h be PartFunc of REAL, REAL n holds
h is_continuous_in x0 iff
(x0 in dom h & for i be Element of NAT st i in Seg n
holds proj(i,n)*h is_continuous_in x0);
theorem :: NFCONT_4:44
for n be non zero Element of NAT, h be PartFunc of REAL, REAL n holds
h is continuous iff
for i be Element of NAT st i in Seg n
holds proj(i,n)*h is continuous;
theorem :: NFCONT_4:45
for x0 be Point of REAL-NS n st 1 <= i & i <= n holds
Proj(i,n) is_continuous_in x0;
theorem :: NFCONT_4:46
for n be non zero Element of NAT,
h be PartFunc of REAL,REAL-NS n holds
h is_continuous_in x0 iff
for i be Element of NAT st i in Seg n
holds Proj(i,n)*h is_continuous_in x0;
theorem :: NFCONT_4:47
for n be non zero Element of NAT,
h be PartFunc of REAL, REAL-NS n holds
h is continuous iff
for i be Element of NAT st i in Seg n
holds Proj(i,n)*h is continuous;