:: deftheorem defines is_convergent_in LIMFUNC3:def 1 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_convergent_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds
( f /* seq is convergent & lim (f /* seq) = g ) ) );