let T be non empty TopSpace; :: thesis: for M being non empty MetrSpace

for f being Function of (TopSpaceMetr M),T holds

( f is continuous iff for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V )

let M be non empty MetrSpace; :: thesis: for f being Function of (TopSpaceMetr M),T holds

( f is continuous iff for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V )

let f be Function of (TopSpaceMetr M),T; :: thesis: ( f is continuous iff for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V )

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V ; :: thesis: f is continuous

for p being Point of (TopSpaceMetr M)

for V being Subset of T st f . p in V & V is open holds

ex W being Subset of (TopSpaceMetr M) st

( p in W & W is open & f .: W c= V )

for f being Function of (TopSpaceMetr M),T holds

( f is continuous iff for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V )

let M be non empty MetrSpace; :: thesis: for f being Function of (TopSpaceMetr M),T holds

( f is continuous iff for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V )

let f be Function of (TopSpaceMetr M),T; :: thesis: ( f is continuous iff for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V )

hereby :: thesis: ( ( for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V ) implies f is continuous )

assume A7:
for p being Point of Mfor V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V ) implies f is continuous )

assume A1:
f is continuous
; :: thesis: for p being Point of M

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V

let p be Point of M; :: thesis: for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V

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

assume f . p in V ; :: thesis: ex s being positive Real st f .: (Ball (p,s)) c= V

then consider W being Subset of (TopSpaceMetr M) such that

A2: p in W and

A3: W is open and

A4: f .: W c= V by A1, JGRAPH_2:10;

Int W = W by A3, TOPS_1:23;

then consider s being Real such that

A5: s > 0 and

A6: Ball (p,s) c= W by A2, GOBOARD6:4;

reconsider s = s as positive Real by A5;

take s = s; :: thesis: f .: (Ball (p,s)) c= V

f .: (Ball (p,s)) c= f .: W by A6, RELAT_1:123;

hence f .: (Ball (p,s)) c= V by A4; :: thesis: verum

end;for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V

let p be Point of M; :: thesis: for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V

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

assume f . p in V ; :: thesis: ex s being positive Real st f .: (Ball (p,s)) c= V

then consider W being Subset of (TopSpaceMetr M) such that

A2: p in W and

A3: W is open and

A4: f .: W c= V by A1, JGRAPH_2:10;

Int W = W by A3, TOPS_1:23;

then consider s being Real such that

A5: s > 0 and

A6: Ball (p,s) c= W by A2, GOBOARD6:4;

reconsider s = s as positive Real by A5;

take s = s; :: thesis: f .: (Ball (p,s)) c= V

f .: (Ball (p,s)) c= f .: W by A6, RELAT_1:123;

hence f .: (Ball (p,s)) c= V by A4; :: thesis: verum

for V being open Subset of T st f . p in V holds

ex s being positive Real st f .: (Ball (p,s)) c= V ; :: thesis: f is continuous

for p being Point of (TopSpaceMetr M)

for V being Subset of T st f . p in V & V is open holds

ex W being Subset of (TopSpaceMetr M) st

( p in W & W is open & f .: W c= V )

proof

hence
f is continuous
by JGRAPH_2:10; :: thesis: verum
let p be Point of (TopSpaceMetr M); :: thesis: for V being Subset of T st f . p in V & V is open holds

ex W being Subset of (TopSpaceMetr M) 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 (TopSpaceMetr M) 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 (TopSpaceMetr M) st

( p in W & W is open & f .: W c= V )

reconsider u = p as Point of M ;

consider s being positive Real such that

A10: f .: (Ball (u,s)) c= V by A7, A8, A9;

reconsider W = Ball (u,s) as Subset of (TopSpaceMetr M) ;

take W ; :: thesis: ( p in W & W is open & f .: W c= V )

thus p in W by GOBOARD6:1; :: thesis: ( W is open & f .: W c= V )

thus W is open by TOPMETR:14; :: thesis: f .: W c= V

thus f .: W c= V by A10; :: thesis: verum

end;ex W being Subset of (TopSpaceMetr M) 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 (TopSpaceMetr M) 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 (TopSpaceMetr M) st

( p in W & W is open & f .: W c= V )

reconsider u = p as Point of M ;

consider s being positive Real such that

A10: f .: (Ball (u,s)) c= V by A7, A8, A9;

reconsider W = Ball (u,s) as Subset of (TopSpaceMetr M) ;

take W ; :: thesis: ( p in W & W is open & f .: W c= V )

thus p in W by GOBOARD6:1; :: thesis: ( W is open & f .: W c= V )

thus W is open by TOPMETR:14; :: thesis: f .: W c= V

thus f .: W c= V by A10; :: thesis: verum