let f be PartFunc of REAL , REAL ; :: thesis: ( f is Lipschitzian implies f is continuous )
set X = dom f;
assume f is Lipschitzian ; :: thesis: f is continuous
then consider r being real number such that
A2: ( 0 < r & ( for x1, x2 being real number st x1 in dom f & x2 in dom f holds
abs ((f . x1) - (f . x2)) <= r * (abs (x1 - x2)) ) ) by XDef3;

now

let x0 be real number ; :: thesis: ( x0 in dom f implies f is_continuous_in x0 )
assume A5: x0 in dom f ; :: thesis: f is_continuous_in x0
for r being real number st 0 < r holds
ex s being real number st
( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) )

proof

let g be real number ; :: thesis: ( 0 < g implies ex s being real number st
( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < g ) ) )
assume A6: 0 < g ; :: thesis: ex s being real number st
( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < g ) )
set s = g / r;
take s' = g / r; :: thesis: ( 0 < s' & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s' holds
abs ((f . x1) - (f . x0)) < g ) )
A7: 0 < r " by A2;
A8: s' = g * (r " ) by XCMPLX_0:def 9;

now

let x1 be real number ; :: thesis: ( x1 in dom f & abs (x1 - x0) < g / r implies abs ((f . x1) - (f . x0)) < g )
assume A9: ( x1 in dom f & abs (x1 - x0) < g / r ) ; :: thesis: abs ((f . x1) - (f . x0)) < g
then r * (abs (x1 - x0)) < (g / r) * r by A2, XREAL_1:70;
then A10: r * (abs (x1 - x0)) < g by A2, XCMPLX_1:88;
abs ((f . x1) - (f . x0)) <= r * (abs (x1 - x0)) by A5, A9, A2;
then abs ((f . x1) - (f . x0)) < g by A10, XXREAL_0:2;
then abs ((f . x1) - (f . x0)) < g ;
hence abs ((f . x1) - (f . x0)) < g ; :: thesis: verum

end;

hence ( 0 < s' & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s' holds
abs ((f . x1) - (f . x0)) < g ) ) by A6, A7, A8, XREAL_1:131; :: thesis: verum

end;

hence f is_continuous_in x0 by Th3; :: thesis: verum

end;

hence f is continuous by Def2; :: thesis: verum