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 ) )

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

hence
f | X is uniformly_continuous
by Th1; :: thesis: verum
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;( 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