let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S
for x0 being Real holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )

let f be PartFunc of REAL, the carrier of S; :: thesis: for x0 being Real holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )

let x0 be Real; :: thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )

thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) ) :: thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; :: thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) )

hence x0 in dom f ; :: thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1

let N01 be Neighbourhood of f /. x0; :: thesis: ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N01

consider r being Real such that
A2: 0 < r and
A3: { p where p is Point of S : ||.(p - (f /. x0)).|| < r } c= N01 by NFCONT_1:def 1;
set N1 = { p where p is Point of S : ||.(p - (f /. x0)).|| < r } ;
consider s being Real such that
A4: 0 < s and
A5: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A1, A2, Th8;
reconsider N = ].(x0 - s),(x0 + s).[ as Neighbourhood of x0 by ;
take N ; :: thesis: for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N01

let x1 be Real; :: thesis: ( x1 in dom f & x1 in N implies f /. x1 in N01 )
assume that
A6: x1 in dom f and
A7: x1 in N ; :: thesis: f /. x1 in N01
|.(x1 - x0).| < s by ;
then ||.((f /. x1) - (f /. x0)).|| < r by A5, A6;
then f /. x1 in { p where p is Point of S : ||.(p - (f /. x0)).|| < r } ;
hence f /. x1 in N01 by A3; :: thesis: verum
end;
assume A8: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) ; :: thesis:
now :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )

assume 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )

then reconsider N1 = { p where p is Point of S : ||.(p - (f /. x0)).|| < r } as Neighbourhood of f /. x0 by NFCONT_1:3;
consider N2 being Neighbourhood of x0 such that
A9: for x1 being Real st x1 in dom f & x1 in N2 holds
f /. x1 in N1 by A8;
consider s being Real such that
A10: 0 < s and
A11: N2 = ].(x0 - s),(x0 + s).[ by RCOMP_1:def 6;
take s = s; :: thesis: ( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )

for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r
proof
let x1 be Real; :: thesis: ( x1 in dom f & |.(x1 - x0).| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A12: x1 in dom f and
A13: |.(x1 - x0).| < s ; :: thesis: ||.((f /. x1) - (f /. x0)).|| < r
x1 in N2 by ;
then f /. x1 in N1 by ;
then ex p being Point of S st
( p = f /. x1 & ||.(p - (f /. x0)).|| < r ) ;
hence ||.((f /. x1) - (f /. x0)).|| < r ; :: thesis: verum
end;
hence ( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) by A10; :: thesis: verum
end;
hence f is_continuous_in x0 by ; :: thesis: verum