let X, Y be non empty TopSpace; :: thesis: for f being Function of X,Y
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let f be Function of X,Y; :: thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X = X1 union X2 implies for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )

assume A1: X = X1 union X2 ; :: thesis: for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x be Point of X; :: thesis: for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x1 be Point of X1; :: thesis: for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x2 be Point of X2; :: thesis: ( x = x1 & x = x2 implies ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )
assume that
A2: x = x1 and
A3: x = x2 ; :: thesis: ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
thus ( f is_continuous_at x implies ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) by A2, A3, Th58; :: thesis: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x )
thus ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x ) :: thesis: verum
proof
assume that
A4: f | X1 is_continuous_at x1 and
A5: f | X2 is_continuous_at x2 ; :: thesis:
for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G
proof
let G be a_neighborhood of f . x; :: thesis: ex H being a_neighborhood of x st f .: H c= G
f . x = (f | X1) . x1 by ;
then consider H1 being a_neighborhood of x1 such that
A6: (f | X1) .: H1 c= G by A4;
the carrier of X1 c= the carrier of X by BORSUK_1:1;
then reconsider S1 = H1 as Subset of X by XBOOLE_1:1;
f . x = (f | X2) . x2 by ;
then consider H2 being a_neighborhood of x2 such that
A7: (f | X2) .: H2 c= G by A5;
the carrier of X2 c= the carrier of X by BORSUK_1:1;
then reconsider S2 = H2 as Subset of X by XBOOLE_1:1;
f .: S2 c= G by ;
then A8: S2 c= f " G by FUNCT_2:95;
consider H being a_neighborhood of x such that
A9: H c= H1 \/ H2 by A1, A2, A3, Th16;
take H ; :: thesis: f .: H c= G
f .: S1 c= G by ;
then S1 c= f " G by FUNCT_2:95;
then S1 \/ S2 c= f " G by ;
then H c= f " G by ;
hence f .: H c= G by FUNCT_2:95; :: thesis: verum
end;
hence f is_continuous_at x ; :: thesis: verum
end;