let f be PartFunc of REAL,REAL; :: thesis: ( ex r being Real st rng f = {r} implies f is continuous )
given r being Real such that A1: rng f = {r} ; :: thesis: f is continuous
now :: thesis: for x1, x2 being Real st x1 in dom f & x2 in dom f holds
|.((f . x1) - (f . x2)).| <= 1 * |.(x1 - x2).|
let x1, x2 be Real; :: thesis: ( x1 in dom f & x2 in dom f implies |.((f . x1) - (f . x2)).| <= 1 * |.(x1 - x2).| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; :: thesis: |.((f . x1) - (f . x2)).| <= 1 * |.(x1 - x2).|
f . x2 in rng f by A3, FUNCT_1:def 3;
then A4: f . x2 = r by A1, TARSKI:def 1;
f . x1 in rng f by A2, FUNCT_1:def 3;
then f . x1 = r by A1, TARSKI:def 1;
then |.((f . x1) - (f . x2)).| = 0 by A4, ABSVALUE:2;
hence |.((f . x1) - (f . x2)).| <= 1 * |.(x1 - x2).| by COMPLEX1:46; :: thesis: verum
end;
then f is Lipschitzian ;
hence f is continuous ; :: thesis: verum