for x0 being Real
for f being PartFunc of REAL,REAL st ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) & ( 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 & f . g1 <> 0 & f . g2 <> 0 ) ) holds
( f ^ is_convergent_in x0 & lim ((f ^),x0) = 0 )