let X, Y be non empty TopSpace; for f being Function of X,Y
for X1, X2 being non empty closed SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; for X1, X2 being non empty closed SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty closed SubSpace of X; ( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated
by TSEP_1:80;
hence
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
by Th117; verum