let T be non empty TopSpace; :: thesis: for M being non empty MetrSpace
for f being Function of T,() holds
( f is continuous iff for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) )

let M be non empty MetrSpace; :: thesis: for f being Function of T,() holds
( f is continuous iff for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) )

let f be Function of T,(); :: thesis: ( f is continuous iff for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) )

thus ( f is continuous implies for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) ) :: thesis: ( ( for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) ) implies f is continuous )
proof
assume A1: f is continuous ; :: thesis: for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) )

let p be Point of T; :: thesis: for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) )

let q be Point of M; :: thesis: for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) )

let r be positive Real; :: thesis: ( q = f . p implies ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) )

assume A2: f . p = q ; :: thesis: ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) )

reconsider V = Ball (q,r) as Subset of () ;
A3: q in Ball (q,r) by GOBOARD6:1;
V is open by TOPMETR:14;
then ex W being Subset of T st
( p in W & W is open & f .: W c= V ) by ;
hence ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) ; :: thesis: verum
end;
assume A4: for p being Point of T
for q being Point of M
for r being positive Real st q = f . p holds
ex W being open Subset of T st
( p in W & f .: W c= Ball (q,r) ) ; :: thesis: f is continuous
for p being Point of T
for V being Subset of () 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 () 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 (); :: 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 A5: 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 ) )

reconsider u = f . p as Point of M ;
assume V is open ; :: thesis: ex W being Subset of T st
( p in W & W is open & f .: W c= V )

then Int V = V by TOPS_1:23;
then consider e being Real such that
A6: ( e > 0 & Ball (u,e) c= V ) by ;
ex W being open Subset of T st
( p in W & f .: W c= Ball (u,e) ) by A4, A6;
hence ex W being Subset of T st
( p in W & W is open & f .: W c= V ) by ; :: thesis: verum
end;
hence f is continuous by JGRAPH_2:10; :: thesis: verum