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

proof

assume A1: f is continuous ; :: thesis: for p being Point of T
for r being positive Real ex W being open Subset of T st
( p in W & f .: W c= ].((f . p) - r),((f . p) + r).[ )
let p be Point of T; :: thesis: for r being positive Real ex W being open Subset of T st
( p in W & f .: W c= ].((f . p) - r),((f . p) + r).[ )
let r be positive Real; :: thesis: ex W being open Subset of T st
( p in W & f .: W c= ].((f . p) - r),((f . p) + r).[ )
A2: f . p in ].((f . p) - r),((f . p) + r).[ by TOPREAL6:15;
R^1 ].((f . p) - r),((f . p) + r).[ is open ;
then ex W being Subset of T st
( p in W & W is open & f .: W c= ].((f . p) - r),((f . p) + r).[ ) by A1, A2, JGRAPH_2:10;
hence ex W being open Subset of T st
( p in W & f .: W c= ].((f . p) - r),((f . p) + r).[ ) ; :: thesis: verum

end;

assume A3: for p being Point of T
for r being positive Real ex W being open Subset of T st
( p in W & f .: W c= ].((f . p) - r),((f . p) + r).[ ) ; :: thesis: f is continuous
for p being Point of T
for V being Subset of R^1 st f . p in V & V is open holds
ex W being Subset of T st
( p in W & W is open & f .: W c= V )

proof

let p be Point of T; :: thesis: for V being Subset of R^1 st f . p in V & V is open holds
ex W being Subset of T st
( p in W & W is open & f .: W c= V )
let V be Subset of R^1; :: thesis: ( f . p in V & V is open implies ex W being Subset of T st
( p in W & W is open & f .: W c= V ) )
assume A4: f . p in V ; :: thesis: ( not V is open or ex W being Subset of T st
( p in W & W is open & f .: W c= V ) )
assume V is open ; :: thesis: ex W being Subset of T st
( p in W & W is open & f .: W c= V )
then consider r being Real such that
A5: r > 0 and
A6: ].((f . p) - r),((f . p) + r).[ c= V by A4, FRECHET:8;
ex W being open Subset of T st
( p in W & f .: W c= ].((f . p) - r),((f . p) + r).[ ) by A3, A5;
hence ex W being Subset of T st
( p in W & W is open & f .: W c= V ) by A6, XBOOLE_1:1; :: thesis: verum

end;

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