let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL holds
( f is_differentiable_on Z iff ( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_on Z iff ( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) ) )
thus ( f is_differentiable_on Z implies ( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) ) ) :: thesis: ( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) implies f is_differentiable_on Z )
proof
assume A1: f is_differentiable_on Z ; :: thesis: ( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) )
hence Z c= dom f ; :: thesis: for x being Real st x in Z holds
f is_differentiable_in x
let x0 be Real; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )
assume A2: x0 in Z ; :: thesis: f is_differentiable_in x0
then f | Z is_differentiable_in x0 by A1;
then consider N being Neighbourhood of x0 such that
A3: N c= dom (f | Z) and
A4: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) ;
take N ; :: according to FDIFF_1:def 4 :: thesis: ( N c= dom f & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )
dom (f | Z) = (dom f) /\ Z by RELAT_1:61;
then dom (f | Z) c= dom f by XBOOLE_1:17;
hence N c= dom f by A3; :: thesis: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
consider L being LinearFunc, R being RestFunc such that
A5: for x being Real st x in N holds
((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A4;
take L ; :: thesis: ex R being RestFunc st
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
take R ; :: thesis: for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
let x be Real; :: thesis: ( x in N implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume A6: x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
then ((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A5;
then (f . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A3, A6, FUNCT_1:47;
hence (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) by A2, FUNCT_1:49; :: thesis: verum
end;
assume that
A7: Z c= dom f and
A8: for x being Real st x in Z holds
f is_differentiable_in x ; :: thesis: f is_differentiable_on Z
thus Z c= dom f by A7; :: 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 A9: x0 in Z ; :: thesis: f | Z is_differentiable_in x0
then consider N1 being Neighbourhood of x0 such that
A10: N1 c= Z by RCOMP_1:18;
f is_differentiable_in x0 by A8, A9;
then consider N being Neighbourhood of x0 such that
A11: N c= dom f and
A12: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ;
consider N2 being Neighbourhood of x0 such that
A13: N2 c= N1 and
A14: N2 c= N by RCOMP_1:17;
A15: N2 c= Z by A10, A13;
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)) )
N2 c= dom f by A11, A14;
then N2 c= (dom f) /\ Z by A15, XBOOLE_1:19;
hence A16: 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
A17: for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) by A12;
A18: x0 in N2 by RCOMP_1:16;
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 A19: x in N2 ; :: thesis: ((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0))
then (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) by A14, A17;
then ((f | Z) . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) by A16, A19, FUNCT_1:47;
hence ((f | Z) . x) - ((f | Z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A16, A18, FUNCT_1:47; :: thesis: verum