let X, Y be non empty TopSpace; :: thesis:
let X1, X2 be non empty SubSpace of X; :: thesis:
let g be Function of (X1 union X2),Y; :: thesis:
let x1 be Point of X1; :: thesis:
let x2 be Point of X2; :: thesis:
let x be Point of (X1 union X2); :: thesis:
assume that
A1: x = x1 and
A2: x = x2 ; :: thesis:
A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;
A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;
hence ( g is_continuous_at x implies ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) by A1, A2, A3, Th74; :: thesis:
thus ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x ) :: thesis:

proof

assume that
A5: g | X1 is_continuous_at x1 and
A6: g | X2 is_continuous_at x2 ; :: thesis:
for G being a_neighborhood of g . x ex H being a_neighborhood of x st g .: H c= G

proof

let G be a_neighborhood of g . x; :: thesis:
g . x = (g | X1) . x1 by A1, A4, Th65;
then consider H1 being a_neighborhood of x1 such that
A7: (g | X1) .: H1 c= G by A5;
g . x = (g | X2) . x2 by A2, A3, Th65;
then consider H2 being a_neighborhood of x2 such that
A8: (g | X2) .: H2 c= G by A6;
the carrier of X2 c= the carrier of (X1 union X2) by A3, TSEP_1:4;
then reconsider S2 = H2 as Subset of (X1 union X2) by XBOOLE_1:1;
g .: S2 c= G by A3, A8, Th68;
then A9: S2 c= g " G by FUNCT_2:95;
the carrier of X1 c= the carrier of (X1 union X2) by A4, TSEP_1:4;
then reconsider S1 = H1 as Subset of (X1 union X2) by XBOOLE_1:1;
consider H being a_neighborhood of x such that
A10: H c= H1 \/ H2 by A1, A2, Th16;
take H ; :: thesis:
g .: S1 c= G by A4, A7, Th68;
then S1 c= g " G by FUNCT_2:95;
then S1 \/ S2 c= g " G by A9, XBOOLE_1:8;
then H c= g " G by A10, XBOOLE_1:1;
hence g .: H c= G by FUNCT_2:95; :: thesis:

end;

hence g is_continuous_at x ; :: thesis:

end;