theorem Th8: :: LIMFUNC3:8

for x0 being Real
for f being PartFunc of REAL,REAL holds
( ( 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 ) ) iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) ) )