for X, Y being non empty TopSpace
for p0 being Point of X
for D being non empty Subset of X
for E being non empty Subset of Y
for f being Function of X,Y st D ` = {p0} & E ` = {(f . p0)} & X is T_2 & Y is T_2 & ( for p being Point of (X | D) holds f . p <> f . p0 ) & f | D is continuous Function of (X | D),(Y | E) & ( for V being Subset of Y st f . p0 in V & V is open holds
ex W being Subset of X st
( p0 in W & W is open & f .: W c= V ) ) holds
f is continuous