let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S st f is total & ( for x1, x2 being Real holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Real st f is_continuous_in x0 holds

f | REAL is continuous

let f be PartFunc of REAL, the carrier of S; :: thesis: ( f is total & ( for x1, x2 being Real holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Real st f is_continuous_in x0 implies f | REAL is continuous )

assume that

A1: f is total and

A2: for x1, x2 being Real holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; :: thesis: ( for x0 being Real holds not f is_continuous_in x0 or f | REAL is continuous )

A3: dom f = REAL by A1, PARTFUN1:def 2;

given x0 being Real such that A4: f is_continuous_in x0 ; :: thesis: f | REAL is continuous

reconsider z0 = 0 as Real ;

(f /. z0) + (f /. z0) = f /. (z0 + z0) by A2;

then (f /. z0) + ((f /. z0) - (f /. z0)) = (f /. z0) - (f /. z0) by RLVECT_1:28;

then (f /. z0) + (0. S) = (f /. z0) - (f /. z0) by RLVECT_1:15;

then A5: (f /. z0) + (0. S) = 0. S by RLVECT_1:15;

f | REAL is continuous

let f be PartFunc of REAL, the carrier of S; :: thesis: ( f is total & ( for x1, x2 being Real holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Real st f is_continuous_in x0 implies f | REAL is continuous )

assume that

A1: f is total and

A2: for x1, x2 being Real holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; :: thesis: ( for x0 being Real holds not f is_continuous_in x0 or f | REAL is continuous )

A3: dom f = REAL by A1, PARTFUN1:def 2;

given x0 being Real such that A4: f is_continuous_in x0 ; :: thesis: f | REAL is continuous

reconsider z0 = 0 as Real ;

(f /. z0) + (f /. z0) = f /. (z0 + z0) by A2;

then (f /. z0) + ((f /. z0) - (f /. z0)) = (f /. z0) - (f /. z0) by RLVECT_1:28;

then (f /. z0) + (0. S) = (f /. z0) - (f /. z0) by RLVECT_1:15;

then A5: (f /. z0) + (0. S) = 0. S by RLVECT_1:15;

A6: now :: thesis: for x1 being Real holds - (f /. x1) = f /. (- x1)

let x1 be Real; :: thesis: - (f /. x1) = f /. (- x1)

0. S = f /. (x1 + (- x1)) by A5, RLVECT_1:4;

then 0. S = (f /. x1) + (f /. (- x1)) by A2;

hence - (f /. x1) = f /. (- x1) by RLVECT_1:def 10; :: thesis: verum

end;0. S = f /. (x1 + (- x1)) by A5, RLVECT_1:4;

then 0. S = (f /. x1) + (f /. (- x1)) by A2;

hence - (f /. x1) = f /. (- x1) by RLVECT_1:def 10; :: thesis: verum

A7: now :: thesis: for x1, x2 being Real holds f /. (x1 - x2) = (f /. x1) - (f /. x2)

let x1, x2 be Real; :: thesis: f /. (x1 - x2) = (f /. x1) - (f /. x2)

f /. (x1 - x2) = f /. (x1 + (- x2)) ;

then f /. (x1 - x2) = (f /. x1) + (f /. (- x2)) by A2;

hence f /. (x1 - x2) = (f /. x1) - (f /. x2) by A6; :: thesis: verum

end;f /. (x1 - x2) = f /. (x1 + (- x2)) ;

then f /. (x1 - x2) = (f /. x1) + (f /. (- x2)) by A2;

hence f /. (x1 - x2) = (f /. x1) - (f /. x2) by A6; :: thesis: verum

now :: thesis: for x1, r being Real st x1 in REAL & r > 0 holds

ex s being Real st

( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) )

hence
f | REAL is continuous
by A3, Th17; :: thesis: verumex s being Real st

( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) )

let x1, r be Real; :: thesis: ( x1 in REAL & r > 0 implies ex s being Real st

( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) ) )

assume that

x1 in REAL and

A8: r > 0 ; :: thesis: ex s being Real st

( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) )

set y = x1 - x0;

consider s being Real such that

A9: 0 < s and

A10: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th8;

take s = s; :: thesis: ( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) )

thus s > 0 by A9; :: thesis: for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r

let x2 be Real; :: thesis: ( x2 in REAL & |.(x2 - x1).| < s implies ||.((f /. x2) - (f /. x1)).|| < r )

assume that

x2 in REAL and

A11: |.(x2 - x1).| < s ; :: thesis: ||.((f /. x2) - (f /. x1)).|| < r

A12: ( x2 - (x1 - x0) in REAL & |.((x2 - (x1 - x0)) - x0).| = |.(x2 - x1).| ) by XREAL_0:def 1;

(x1 - x0) + x0 = x1 ;

then ||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2

.= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27

.= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ;

hence ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A12; :: thesis: verum

end;( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) ) )

assume that

x1 in REAL and

A8: r > 0 ; :: thesis: ex s being Real st

( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) )

set y = x1 - x0;

consider s being Real such that

A9: 0 < s and

A10: for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds

||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th8;

take s = s; :: thesis: ( s > 0 & ( for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r ) )

thus s > 0 by A9; :: thesis: for x2 being Real st x2 in REAL & |.(x2 - x1).| < s holds

||.((f /. x2) - (f /. x1)).|| < r

let x2 be Real; :: thesis: ( x2 in REAL & |.(x2 - x1).| < s implies ||.((f /. x2) - (f /. x1)).|| < r )

assume that

x2 in REAL and

A11: |.(x2 - x1).| < s ; :: thesis: ||.((f /. x2) - (f /. x1)).|| < r

A12: ( x2 - (x1 - x0) in REAL & |.((x2 - (x1 - x0)) - x0).| = |.(x2 - x1).| ) by XREAL_0:def 1;

(x1 - x0) + x0 = x1 ;

then ||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2

.= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27

.= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ;

hence ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A12; :: thesis: verum