let X be set ; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_on X & Z c= X implies f is_differentiable_on Z )
assume that
A1: f is_differentiable_on X and
A2: Z c= X ; :: thesis: f is_differentiable_on Z
X c= dom f by A1;
hence Z c= dom f by A2; :: according to FDIFF_1:def 6 :: thesis: for x being Real st x in Z holds
f | Z is_differentiable_in x
let x0 be Real; :: thesis: ( x0 in Z implies f | Z is_differentiable_in x0 )
assume A3: x0 in Z ; :: thesis: f | Z is_differentiable_in x0
then f | X is_differentiable_in x0 by A1, A2;
then consider N being Neighbourhood of x0 such that
A4: N c= dom (f | X) and
A5: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((f | X) . x) - ((f | X) . x0) = (L . (x - x0)) + (R . (x - x0)) ;
consider N1 being Neighbourhood of x0 such that
A6: N1 c= Z by A3, RCOMP_1:18;
consider N2 being Neighbourhood of x0 such that
A7: N2 c= N and
A8: N2 c= N1 by RCOMP_1:17;
A9: N2 c= Z by A6, A8;
take N2 ; :: according to FDIFF_1:def 4 :: thesis: ( N2 c= dom (f | Z) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N2 holds
((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) )
dom (f | X) = (dom f) /\ X by RELAT_1:61;
then dom (f | X) c= dom f by XBOOLE_1:17;
then N c= dom f by A4;
then N2 c= dom f by A7;
then N2 c= (dom f) /\ Z by A9, XBOOLE_1:19;
hence A10: N2 c= dom (f | Z) by RELAT_1:61; :: thesis: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N2 holds
((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0))
consider L being LinearFunc, R being RestFunc such that
A11: for x being Real st x in N holds
((f | X) . x) - ((f | X) . x0) = (L . (x - x0)) + (R . (x - x0)) by A5;
take L ; :: thesis: ex R being RestFunc st
for x being Real st x in N2 holds
((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0))
take R ; :: thesis: for x being Real st x in N2 holds
((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0))
let x be Real; :: thesis: ( x in N2 implies ((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume A12: x in N2 ; :: thesis: ((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0))
then ((f | X) . x) - ((f | X) . x0) = (L . (x - x0)) + (R . (x - x0)) by A7, A11;
then A13: ((f | X) . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) by A2, A3, FUNCT_1:49;
x in N by A7, A12;
then (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) by A4, A13, FUNCT_1:47;
then (f . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A3, FUNCT_1:49;
hence ((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A10, A12, FUNCT_1:47; :: thesis: verum