let x0 be Real; :: thesis: for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0

let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S st ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0

let f be PartFunc of REAL, the carrier of S; :: thesis: ( ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} implies f is_continuous_in x0 )
given N being Neighbourhood of x0 such that A1: (dom f) /\ N = {x0} ; :: thesis:
x0 in (dom f) /\ N by ;
then A2: x0 in dom f by XBOOLE_0:def 4;
now :: thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1
let N1 be Neighbourhood of f /. x0; :: thesis: ex N being Neighbourhood of x0 st f .: N c= N1
take N = N; :: thesis: f .: N c= N1
A3: f /. x0 in N1 by NFCONT_1:4;
f .: N = Im (f,x0) by
.= {(f . x0)} by
.= {(f /. x0)} by ;
hence f .: N c= N1 by ; :: thesis: verum
end;
hence f is_continuous_in x0 by ; :: thesis: verum