let X be set ; :: thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
f is_uniformly_continuous_on X
let S, T be RealNormSpace; :: thesis: for f being PartFunc of S,T st f is_Lipschitzian_on X holds
f is_uniformly_continuous_on X
let f be PartFunc of S,T; :: thesis: ( f is_Lipschitzian_on X implies f is_uniformly_continuous_on X )
assume A1: f is_Lipschitzian_on X ; :: thesis: f is_uniformly_continuous_on X
hence X c= dom f by NFCONT_1:def 13; :: according to NFCONT_2:def 1 :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r ) )
consider r being Real such that
A2: ( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) by A1, NFCONT_1:def 13;
let p be Real; :: thesis: ( 0 < p implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in 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 Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p ) )
take s = p / r; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p ) )
thus 0 < s by A2, A3, XREAL_1:141; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p
let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.((f /. x1) - (f /. x2)).|| < p )
assume A4: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| < p
then A5: r * ||.(x1 - x2).|| < s * r by A2, XREAL_1:70;
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by A2, A4;
then ||.((f /. x1) - (f /. x2)).|| < (p / r) * r by A5, XXREAL_0:2;
hence ||.((f /. x1) - (f /. x2)).|| < p by A2, XCMPLX_1:88; :: thesis: verum