:: deftheorem defines is_continuous_in DUALSP04:def 5 :
for X being RealUnitarySpace
for f being PartFunc of the carrier of X,REAL
for x0 being Point of X holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of X st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) );