theorem :: TOPREALA:18

for f, f1, f2 being PartFunc of REAL,REAL st dom f = (dom f1) \/ (dom f2) & dom f1 is open & dom f2 is open & f1 | (dom f1) is continuous & f2 | (dom f2) is continuous & ( for z being set st z in dom f1 holds
f . z = f1 . z ) & ( for z being set st z in dom f2 holds
f . z = f2 . z ) holds
f | (dom f) is continuous