let f be PartFunc of REAL,REAL; :: thesis: ( f is V8() implies f is continuous )

assume A1: f is V8() ; :: thesis: f is continuous

hence f is continuous ; :: thesis: verum

assume A1: f is V8() ; :: thesis: f is continuous

now :: thesis: for x0, r being Real st x0 in dom f & 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 ) )

then
f | (dom f) is continuous
by Th14;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 ) )

reconsider s = 1 as Real ;

let x0, r be Real; :: thesis: ( x0 in dom f & 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 that

A2: x0 in dom f and

A3: 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 ) )

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

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

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

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

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

assume A4: x1 in dom f ; :: thesis: ( |.(x1 - x0).| < s implies |.((f . x1) - (f . x0)).| < r )

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

f . x1 = f . x0 by A1, A2, A4;

hence |.((f . x1) - (f . x0)).| < r by A3, ABSVALUE:2; :: thesis: verum

end;let x0, r be Real; :: thesis: ( x0 in dom f & 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 that

A2: x0 in dom f and

A3: 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 ) )

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

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

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

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

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

assume A4: x1 in dom f ; :: thesis: ( |.(x1 - x0).| < s implies |.((f . x1) - (f . x0)).| < r )

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

f . x1 = f . x0 by A1, A2, A4;

hence |.((f . x1) - (f . x0)).| < r by A3, ABSVALUE:2; :: thesis: verum

hence f is continuous ; :: thesis: verum