let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S st ex r being Point of S st rng f = {r} holds
f is continuous
let f be PartFunc of REAL, the carrier of S; :: thesis: ( ex r being Point of S st rng f = {r} implies f is continuous )
given r being Point of S such that A1: rng f = {r} ; :: thesis: f is continuous

now

let x1, x2 be real number ; :: thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) )
assume A2: ( x1 in dom f & x2 in dom f ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2))
then f . x2 in rng f by FUNCT_1:def 5;
then f /. x2 in rng f by A2, PARTFUN1:def 8;
then A4: f /. x2 = r by A1, TARSKI:def 1;
f . x1 in rng f by A2, FUNCT_1:def 5;
then f /. x1 in rng f by A2, PARTFUN1:def 8;
then f /. x1 = r by A1, TARSKI:def 1;
then ||.((f /. x1) - (f /. x2)).|| = 0 by A4, NORMSP_1:10;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) by COMPLEX1:132; :: thesis: verum

end;

then f is Lipschitzian by Def3;
hence f is continuous ; :: thesis: verum