let X be set ; :: thesis: for f being PartFunc of REAL,REAL st X c= dom f & f | X is uniformly_continuous holds

f | X is continuous

let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & f | X is uniformly_continuous implies f | X is continuous )

assume A1: X c= dom f ; :: thesis: ( not f | X is uniformly_continuous or f | X is continuous )

assume A2: f | X is uniformly_continuous ; :: thesis: f | X is continuous

f | X is continuous

let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & f | X is uniformly_continuous implies f | X is continuous )

assume A1: X c= dom f ; :: thesis: ( not f | X is uniformly_continuous or f | X is continuous )

assume A2: f | X is uniformly_continuous ; :: thesis: f | X is continuous

now :: thesis: for x0, r being Real st x0 in X & 0 < r holds

ex s being Real st

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

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

hence
f | X is continuous
by A1, FCONT_1:14; :: thesis: verumex s being Real st

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

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

let x0, r be Real; :: thesis: ( x0 in X & 0 < r implies ex s being Real st

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

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

assume that

A3: x0 in X and

A4: 0 < r ; :: thesis: ex s being Real st

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

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

A5: x0 in dom (f | X) by A1, A3, RELAT_1:62;

consider s being Real such that

A6: 0 < s and

A7: 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 by A2, A4, Th1;

reconsider s = s as Real ;

take s = s; :: thesis: ( 0 < s & ( for x1 being Real st x1 in X & |.(x1 - x0).| < s holds

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

thus 0 < s by A6; :: thesis: for x1 being Real st x1 in X & |.(x1 - x0).| < s holds

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

let x1 be Real; :: thesis: ( x1 in X & |.(x1 - x0).| < s implies |.((f . x1) - (f . x0)).| < r )

assume that

A8: x1 in X and

A9: |.(x1 - x0).| < s ; :: thesis: |.((f . x1) - (f . x0)).| < r

x1 in dom (f | X) by A1, A8, RELAT_1:62;

hence |.((f . x1) - (f . x0)).| < r by A7, A9, A5; :: thesis: verum

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

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

assume that

A3: x0 in X and

A4: 0 < r ; :: thesis: ex s being Real st

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

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

A5: x0 in dom (f | X) by A1, A3, RELAT_1:62;

consider s being Real such that

A6: 0 < s and

A7: 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 by A2, A4, Th1;

reconsider s = s as Real ;

take s = s; :: thesis: ( 0 < s & ( for x1 being Real st x1 in X & |.(x1 - x0).| < s holds

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

thus 0 < s by A6; :: thesis: for x1 being Real st x1 in X & |.(x1 - x0).| < s holds

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

let x1 be Real; :: thesis: ( x1 in X & |.(x1 - x0).| < s implies |.((f . x1) - (f . x0)).| < r )

assume that

A8: x1 in X and

A9: |.(x1 - x0).| < s ; :: thesis: |.((f . x1) - (f . x0)).| < r

x1 in dom (f | X) by A1, A8, RELAT_1:62;

hence |.((f . x1) - (f . x0)).| < r by A7, A9, A5; :: thesis: verum