let X be set ; for f being PartFunc of REAL,REAL holds
( f | X is Lipschitzian iff ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )
let f be PartFunc of REAL,REAL; ( f | X is Lipschitzian iff ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )
thus
( f | X is Lipschitzian implies ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )
( ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) implies f | X is Lipschitzian )
given r being Real such that A4:
0 < r
and
A5:
for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).|
; f | X is Lipschitzian
take
r
; FCONT_1:def 3 ( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| ) )
thus
0 < r
by A4; for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|
let x1, x2 be Real; ( x1 in dom (f | X) & x2 in dom (f | X) implies |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| )
assume A6:
( x1 in dom (f | X) & x2 in dom (f | X) )
; |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|
then
( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 )
by FUNCT_1:47;
hence
|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|
by A5, A6; verum