let m be Nat; :: thesis:
let f be Function of (TOP-REAL m),R^1; :: thesis:
A1: TopStruct(# the U1 of (TOP-REAL m), the topology of (TOP-REAL m) #) = TopSpaceMetr (Euclid m) by EUCLID:def 8;
then reconsider f1 = f as Function of (TopSpaceMetr (Euclid m)),R^1 ;

hereby :: thesis:

assume A2: f is continuous ; :: thesis:
let p be Point of (TOP-REAL m); :: thesis:
let r be positive Real; :: thesis:
reconsider p1 = p as Point of (Euclid m) by EUCLID:67;
reconsider q1 = f . p as Point of RealSpace ;
f1 is continuous by A1, A2, YELLOW12:36;
then consider s being positive Real such that
A3: f .: (Ball (p1,s)) c= Ball (q1,r) by A1, Th17;
take s = s; :: thesis:
( Ball (p1,s) = Ball (p,s) & Ball (q1,r) = ].((f . p) - r),((f . p) + r).[ ) by FRECHET:7, TOPREAL9:13;
hence f .: (Ball (p,s)) c= ].((f . p) - r),((f . p) + r).[ by A3; :: thesis:

end;

assume A4: for p being Point of (TOP-REAL m)
for r being positive Real ex s being positive Real st f .: (Ball (p,s)) c= ].((f . p) - r),((f . p) + r).[ ; :: thesis:
for p being Point of (Euclid m)
for q being Point of RealSpace
for r being positive Real st q = f1 . p holds
ex s being positive Real st f1 .: (Ball (p,s)) c= Ball (q,r)

proof

let p be Point of (Euclid m); :: thesis:
let q be Point of RealSpace; :: thesis:
let r be positive Real; :: thesis:
assume A5: q = f1 . p ; :: thesis:
reconsider p1 = p as Point of (TOP-REAL m) by EUCLID:67;
consider s being positive Real such that
A6: f .: (Ball (p1,s)) c= ].((f . p) - r),((f . p) + r).[ by A4;
take s ; :: thesis:
( Ball (p1,s) = Ball (p,s) & ].((f . p) - r),((f . p) + r).[ = Ball (q,r) ) by A5, FRECHET:7, TOPREAL9:13;
hence f1 .: (Ball (p,s)) c= Ball (q,r) by A6; :: thesis:

end;

then f1 is continuous by Th17;
hence f is continuous by A1, YELLOW12:36; :: thesis: