let X be set ; :: thesis: for f being PartFunc of REAL ,REAL st X c= dom f & f | X is uniformly_continuous holds
f | X is continuous
let f be PartFunc of REAL ,REAL ; :: thesis: ( X c= dom f & f | X is uniformly_continuous implies f | X is continuous )
assume Z: X c= dom f ; :: thesis: ( not f | X is uniformly_continuous or f | X is continuous )
assume A1: f | X is uniformly_continuous ; :: thesis: f | X is continuous

now

let x0, r be real number ; :: thesis: ( x0 in X & 0 < r implies ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) ) )
A3: ( x0 is Real & r is Real ) by XREAL_0:def 1;
assume A4: ( x0 in X & 0 < r ) ; :: thesis: ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) )
then consider s being Real such that
A5: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) ) by A1, A3, Def1;
reconsider s = s as real number ;
take s = s; :: thesis: ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) )
thus 0 < s by A5; :: thesis: for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r
let x1 be real number ; :: thesis: ( x1 in X & abs (x1 - x0) < s implies abs ((f . x1) - (f . x0)) < r )
assume F: ( x1 in X & abs (x1 - x0) < s ) ; :: thesis: abs ((f . x1) - (f . x0)) < r
then ( x1 in dom (f | X) & x0 in dom (f | X) ) by A4, Z, RELAT_1:91;
hence abs ((f . x1) - (f . x0)) < r by A5, F; :: thesis: verum

end;

hence f | X is continuous by Z, FCONT_1:15; :: thesis: verum