for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )