let X be set ; :: thesis: for f being PartFunc of REAL,REAL st f | X is Lipschitzian holds
f | X is uniformly_continuous
let f be PartFunc of REAL,REAL; :: thesis: ( f | X is Lipschitzian implies f | X is uniformly_continuous )
assume f | X is Lipschitzian ; :: thesis: f | X is uniformly_continuous
then consider r being Real such that
A1: 0 < r and
A2: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| by FCONT_1:32;
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r ) )
proof
let p be Real; :: thesis: ( 0 < p implies ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < p ) ) )
assume A3: 0 < p ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < p ) )
take s = p / r; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < p ) )
thus 0 < s by A1, A3, XREAL_1:139; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < p
let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s implies |.((f . x1) - (f . x2)).| < p )
assume that
A4: x1 in dom (f | X) and
A5: x2 in dom (f | X) and
A6: |.(x1 - x2).| < s ; :: thesis: |.((f . x1) - (f . x2)).| < p
A7: r * |.(x1 - x2).| < s * r by A1, A6, XREAL_1:68;
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| by A2, A4, A5;
then |.((f . x1) - (f . x2)).| < (p / r) * r by A7, XXREAL_0:2;
hence |.((f . x1) - (f . x2)).| < p by A1, XCMPLX_1:87; :: thesis: verum
end;
hence f | X is uniformly_continuous by Th1; :: thesis: verum