let T be non empty TopSpace; :: thesis: for f being Function of R^1,T holds
( f is continuous iff for p being Point of R^1
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: ].(p - s),(p + s).[ c= V )
let f be Function of R^1,T; :: thesis: ( f is continuous iff for p being Point of R^1
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: ].(p - s),(p + s).[ c= V )

hereby :: thesis: ( ( for p being Point of R^1
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: ].(p - s),(p + s).[ c= V ) implies f is continuous )

assume A1: f is continuous ; :: thesis: for p being Point of R^1
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: ].(p - s),(p + s).[ c= V
let p be Point of R^1; :: thesis: for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: ].(p - s),(p + s).[ c= V
let V be open Subset of T; :: thesis: ( f . p in V implies ex s being positive Real st f .: ].(p - s),(p + s).[ c= V )
assume f . p in V ; :: thesis: ex s being positive Real st f .: ].(p - s),(p + s).[ c= V
then consider W being Subset of R^1 such that
A2: p in W and
A3: W is open and
A4: f .: W c= V by A1, JGRAPH_2:10;
consider s being Real such that
A5: s > 0 and
A6: ].(p - s),(p + s).[ c= W by A2, A3, FRECHET:8;
reconsider s = s as positive Real by A5;
take s = s; :: thesis: f .: ].(p - s),(p + s).[ c= V
f .: ].(p - s),(p + s).[ c= f .: W by A6, RELAT_1:123;
hence f .: ].(p - s),(p + s).[ c= V by A4; :: thesis: verum

end;

assume A7: for p being Point of R^1
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: ].(p - s),(p + s).[ c= V ; :: thesis: f is continuous
for p being Point of R^1
for V being Subset of T st f . p in V & V is open holds
ex W being Subset of R^1 st
( p in W & W is open & f .: W c= V )

proof

let p be Point of R^1; :: thesis: for V being Subset of T st f . p in V & V is open holds
ex W being Subset of R^1 st
( p in W & W is open & f .: W c= V )
let V be Subset of T; :: thesis: ( f . p in V & V is open implies ex W being Subset of R^1 st
( p in W & W is open & f .: W c= V ) )
assume that
A8: f . p in V and
A9: V is open ; :: thesis: ex W being Subset of R^1 st
( p in W & W is open & f .: W c= V )
consider s being positive Real such that
A10: f .: ].(p - s),(p + s).[ c= V by A7, A8, A9;
take W = R^1 ].(p - s),(p + s).[; :: thesis: ( p in W & W is open & f .: W c= V )
thus p in W by TOPREAL6:15; :: thesis: ( W is open & f .: W c= V )
thus W is open ; :: thesis: f .: W c= V
thus f .: W c= V by A10; :: thesis: verum

end;

hence f is continuous by JGRAPH_2:10; :: thesis: verum