let m be non zero Element of NAT ; for k being Element of NAT
for X being non empty Subset of (REAL m)
for r being Real
for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds
r (#) f is_continuously_differentiable_up_to_order k,X
let k be Element of NAT ; for X being non empty Subset of (REAL m)
for r being Real
for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds
r (#) f is_continuously_differentiable_up_to_order k,X
let X be non empty Subset of (REAL m); for r being Real
for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds
r (#) f is_continuously_differentiable_up_to_order k,X
let r be Real; for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds
r (#) f is_continuously_differentiable_up_to_order k,X
let f be PartFunc of (REAL m),REAL; ( f is_continuously_differentiable_up_to_order k,X & X is open implies r (#) f is_continuously_differentiable_up_to_order k,X )
assume A1:
( f is_continuously_differentiable_up_to_order k,X & X is open )
; r (#) f is_continuously_differentiable_up_to_order k,X
for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds
(r (#) f) `partial| (X,I) is_continuous_on X
proof
let I be non
empty FinSequence of
NAT ;
( len I <= k & rng I c= Seg m implies (r (#) f) `partial| (X,I) is_continuous_on X )
assume A2:
(
len I <= k &
rng I c= Seg m )
;
(r (#) f) `partial| (X,I) is_continuous_on X
A3:
f is_partial_differentiable_on X,
I
by A2, PDIFF_9:def 11, A1;
reconsider f0 =
f `partial| (
X,
I) as
PartFunc of
(REAL m),
REAL ;
X = dom f0
by Th1, A2, PDIFF_9:def 11, A1;
then
r (#) f0 is_continuous_on X
by A2, A1, PDIFF_9:47;
hence
(r (#) f) `partial| (
X,
I)
is_continuous_on X
by A1, A2, A3, PDIFF_9:79;
verum
end;
hence
r (#) f is_continuously_differentiable_up_to_order k,X
by PDIFF_9:86, A1, VALUED_1:def 5; verum