let f be PartFunc of REAL, the carrier of S; :: thesis: ( f is constant implies f is continuous )
assume A1: f is constant ; :: thesis: f is continuous

now :: thesis: for x0, r being Real st x0 in dom f & 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 ) )

reconsider s = 1 as Real ;
let x0, r be Real; :: thesis: ( x0 in dom f & 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 that
A2: x0 in dom f and
A3: 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 ) )
take s = s; :: thesis: ( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
thus 0 < s ; :: thesis: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r
let x1 be Real; :: thesis: ( x1 in dom f & |.(x1 - x0).| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume A4: x1 in dom f ; :: thesis: ( |.(x1 - x0).| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume |.(x1 - x0).| < s ; :: thesis: ||.((f /. x1) - (f /. x0)).|| < r
f /. x1 = f . x1 by A4, PARTFUN1:def 6
.= f . x0 by A1, A2, A4, FUNCT_1:def 10
.= f /. x0 by A2, PARTFUN1:def 6 ;
hence ||.((f /. x1) - (f /. x0)).|| < r by A3, NORMSP_1:6; :: thesis: verum

end;

then f | (dom f) is continuous by Th17;
hence f is continuous ; :: thesis: verum