let X, Y be non empty TopSpace; for f being Function of X,Y
for X1, X2 being non empty closed SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,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 st X = X1 union X2 holds
( f is continuous Function of X,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; ( X = X1 union X2 implies ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) )
assume A1:
X = X1 union X2
; ( f is continuous Function of X,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 is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
by A1, Th120; verum