let T be non empty TopSpace; :: thesis: for M being non empty MetrSpace
for f being Function of T,(TopSpaceMetr M) 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,(TopSpaceMetr M) 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,(TopSpaceMetr M); :: 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 (TopSpaceMetr M) ;
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 A1, A2, A3, JGRAPH_2:10;
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 (TopSpaceMetr M) 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 (TopSpaceMetr M) 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 (TopSpaceMetr M); :: 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 A5, GOBOARD6:4;
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 A6, XBOOLE_1:1; :: thesis: verum

end;

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