theorem :: LIMFUNC3:9

for x0 being Real
for f being PartFunc of REAL,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 g1 being Real st 0 < g1 holds
ex g2 being Real st
( 0 < g2 & ( for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ) )