let X be set ; 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; ( f | X is Lipschitzian implies f | X is uniformly_continuous )
assume
f | X is Lipschitzian
; 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;
( 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
;
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;
( 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;
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;
( 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
;
|.((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;
verum
end;
hence
f | X is uniformly_continuous
by Th1; verum