let f be PartFunc of REAL,REAL; :: thesis: for x0 being Real holds

( f is_continuous_in x0 iff for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 )

let x0 be Real; :: thesis: ( f is_continuous_in x0 iff for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 )

thus ( f is_continuous_in x0 implies for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 ) :: thesis: ( ( for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 ) implies f is_continuous_in x0 )

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 ; :: thesis: f is_continuous_in x0

( f is_continuous_in x0 iff for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 )

let x0 be Real; :: thesis: ( f is_continuous_in x0 iff for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 )

thus ( f is_continuous_in x0 implies for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 ) :: thesis: ( ( for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 ) implies f is_continuous_in x0 )

proof

assume A8:
for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st
assume A1:
f is_continuous_in x0
; :: thesis: for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1

let N1 be Neighbourhood of f . x0; :: thesis: ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1

consider r being Real such that

A2: 0 < r and

A3: N1 = ].((f . x0) - r),((f . x0) + r).[ by RCOMP_1:def 6;

consider s being Real such that

A4: 0 < s and

A5: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < r by A1, A2, Th3;

reconsider N = ].(x0 - s),(x0 + s).[ as Neighbourhood of x0 by A4, RCOMP_1:def 6;

take N ; :: thesis: for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1

let x1 be Real; :: thesis: ( x1 in dom f & x1 in N implies f . x1 in N1 )

assume that

A6: x1 in dom f and

A7: x1 in N ; :: thesis: f . x1 in N1

|.(x1 - x0).| < s by A7, RCOMP_1:1;

then |.((f . x1) - (f . x0)).| < r by A5, A6;

hence f . x1 in N1 by A3, RCOMP_1:1; :: thesis: verum

end;for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1

let N1 be Neighbourhood of f . x0; :: thesis: ex N being Neighbourhood of x0 st

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1

consider r being Real such that

A2: 0 < r and

A3: N1 = ].((f . x0) - r),((f . x0) + r).[ by RCOMP_1:def 6;

consider s being Real such that

A4: 0 < s and

A5: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

|.((f . x1) - (f . x0)).| < r by A1, A2, Th3;

reconsider N = ].(x0 - s),(x0 + s).[ as Neighbourhood of x0 by A4, RCOMP_1:def 6;

take N ; :: thesis: for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1

let x1 be Real; :: thesis: ( x1 in dom f & x1 in N implies f . x1 in N1 )

assume that

A6: x1 in dom f and

A7: x1 in N ; :: thesis: f . x1 in N1

|.(x1 - x0).| < s by A7, RCOMP_1:1;

then |.((f . x1) - (f . x0)).| < r by A5, A6;

hence f . x1 in N1 by A3, RCOMP_1:1; :: thesis: verum

for x1 being Real st x1 in dom f & x1 in N holds

f . x1 in N1 ; :: thesis: f is_continuous_in x0

now :: thesis: 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 ) )

hence
f is_continuous_in x0
by Th3; :: thesis: verumex s being Real st

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

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

let r be Real; :: thesis: ( 0 < r 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)).| < r ) ) )

assume 0 < r ; :: 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)).| < r ) )

then reconsider N1 = ].((f . x0) - r),((f . x0) + r).[ as Neighbourhood of f . x0 by RCOMP_1:def 6;

consider N2 being Neighbourhood of x0 such that

A9: for x1 being Real st x1 in dom f & x1 in N2 holds

f . x1 in N1 by A8;

consider s being Real such that

A10: 0 < s and

A11: N2 = ].(x0 - s),(x0 + s).[ by RCOMP_1:def 6;

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

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

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

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

|.((f . x1) - (f . x0)).| < r ) ) by A10; :: thesis: verum

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

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

assume 0 < r ; :: 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)).| < r ) )

then reconsider N1 = ].((f . x0) - r),((f . x0) + r).[ as Neighbourhood of f . x0 by RCOMP_1:def 6;

consider N2 being Neighbourhood of x0 such that

A9: for x1 being Real st x1 in dom f & x1 in N2 holds

f . x1 in N1 by A8;

consider s being Real such that

A10: 0 < s and

A11: N2 = ].(x0 - s),(x0 + s).[ by RCOMP_1:def 6;

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

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

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

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

proof

hence
( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
let x1 be Real; :: thesis: ( x1 in dom f & |.(x1 - x0).| < s implies |.((f . x1) - (f . x0)).| < r )

assume that

A12: x1 in dom f and

A13: |.(x1 - x0).| < s ; :: thesis: |.((f . x1) - (f . x0)).| < r

x1 in N2 by A11, A13, RCOMP_1:1;

then f . x1 in N1 by A9, A12;

hence |.((f . x1) - (f . x0)).| < r by RCOMP_1:1; :: thesis: verum

end;assume that

A12: x1 in dom f and

A13: |.(x1 - x0).| < s ; :: thesis: |.((f . x1) - (f . x0)).| < r

x1 in N2 by A11, A13, RCOMP_1:1;

then f . x1 in N1 by A9, A12;

hence |.((f . x1) - (f . x0)).| < r by RCOMP_1:1; :: thesis: verum

|.((f . x1) - (f . x0)).| < r ) ) by A10; :: thesis: verum