let f be PartFunc of REAL,REAL; :: thesis: ( f is Lipschitzian implies f is continuous )

set X = dom f;

assume f is Lipschitzian ; :: thesis: f is continuous

then consider r being Real such that

A1: 0 < r and

A2: for x1, x2 being Real st x1 in dom f & x2 in dom f holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ;

set X = dom f;

assume f is Lipschitzian ; :: thesis: f is continuous

then consider r being Real such that

A1: 0 < r and

A2: for x1, x2 being Real st x1 in dom f & x2 in dom f holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ;

now :: thesis: for x0 being Real st x0 in dom f holds

f is_continuous_in x0

hence
f is continuous
; :: thesis: verumf is_continuous_in x0

let x0 be Real; :: thesis: ( x0 in dom f implies f is_continuous_in x0 )

assume A3: x0 in dom f ; :: thesis: f is_continuous_in x0

for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < r ) )

end;assume A3: x0 in dom f ; :: thesis: f is_continuous_in x0

for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < r ) )

proof

hence
f is_continuous_in x0
by Th3; :: thesis: verum
let g be Real; :: thesis: ( 0 < g implies ex s being Real st

( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < g ) ) )

assume A4: 0 < g ; :: thesis: ex s being Real st

( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < g ) )

set s = g / r;

take s9 = g / r; :: thesis: ( 0 < s9 & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s9 holds

|.((f . x1) - (f . x0)).| < g ) )

hence ( 0 < s9 & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s9 holds

|.((f . x1) - (f . x0)).| < g ) ) by A1, A4, A5, XREAL_1:129; :: thesis: verum

end;( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < g ) ) )

assume A4: 0 < g ; :: thesis: ex s being Real st

( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < g ) )

set s = g / r;

take s9 = g / r; :: thesis: ( 0 < s9 & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s9 holds

|.((f . x1) - (f . x0)).| < g ) )

A5: now :: thesis: for x1 being Real st x1 in dom f & |.(x1 - x0).| < g / r holds

|.((f . x1) - (f . x0)).| < g

s9 = g * (r ")
by XCMPLX_0:def 9;|.((f . x1) - (f . x0)).| < g

let x1 be Real; :: thesis: ( x1 in dom f & |.(x1 - x0).| < g / r implies |.((f . x1) - (f . x0)).| < g )

assume that

A6: x1 in dom f and

A7: |.(x1 - x0).| < g / r ; :: thesis: |.((f . x1) - (f . x0)).| < g

r * |.(x1 - x0).| < (g / r) * r by A1, A7, XREAL_1:68;

then A8: r * |.(x1 - x0).| < g by A1, XCMPLX_1:87;

|.((f . x1) - (f . x0)).| <= r * |.(x1 - x0).| by A2, A3, A6;

hence |.((f . x1) - (f . x0)).| < g by A8, XXREAL_0:2; :: thesis: verum

end;assume that

A6: x1 in dom f and

A7: |.(x1 - x0).| < g / r ; :: thesis: |.((f . x1) - (f . x0)).| < g

r * |.(x1 - x0).| < (g / r) * r by A1, A7, XREAL_1:68;

then A8: r * |.(x1 - x0).| < g by A1, XCMPLX_1:87;

|.((f . x1) - (f . x0)).| <= r * |.(x1 - x0).| by A2, A3, A6;

hence |.((f . x1) - (f . x0)).| < g by A8, XXREAL_0:2; :: thesis: verum

hence ( 0 < s9 & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s9 holds

|.((f . x1) - (f . x0)).| < g ) ) by A1, A4, A5, XREAL_1:129; :: thesis: verum