let Y be Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st f is_differentiable_on Y holds

Y is open

let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_on Y implies Y is open )

assume A1: f is_differentiable_on Y ; :: thesis: Y is open

Y is open

let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_on Y implies Y is open )

assume A1: f is_differentiable_on Y ; :: thesis: Y is open

now :: thesis: for x0 being Element of REAL st x0 in Y holds

ex N being Neighbourhood of x0 st N c= Y

hence
Y is open
by RCOMP_1:20; :: thesis: verumex N being Neighbourhood of x0 st N c= Y

let x0 be Element of REAL ; :: thesis: ( x0 in Y implies ex N being Neighbourhood of x0 st N c= Y )

assume x0 in Y ; :: thesis: ex N being Neighbourhood of x0 st N c= Y

then f | Y is_differentiable_in x0 by A1;

then consider N being Neighbourhood of x0 such that

A2: N c= dom (f | Y) and

ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

((f | Y) . x) - ((f | Y) . x0) = (L . (x - x0)) + (R . (x - x0)) ;

take N = N; :: thesis: N c= Y

thus N c= Y by A2, XBOOLE_1:1; :: thesis: verum

end;assume x0 in Y ; :: thesis: ex N being Neighbourhood of x0 st N c= Y

then f | Y is_differentiable_in x0 by A1;

then consider N being Neighbourhood of x0 such that

A2: N c= dom (f | Y) and

ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

((f | Y) . x) - ((f | Y) . x0) = (L . (x - x0)) + (R . (x - x0)) ;

take N = N; :: thesis: N c= Y

thus N c= Y by A2, XBOOLE_1:1; :: thesis: verum