Approximate continuity of a real-valued measurable function 
on 
at 
means that for every 
,
 |
where 
is the Lebesgue measure of the measurable
set 
. Every continuous
function is approximately continuous, and every measurable
function is approximately continuous almost
everywhere (Ciesielski et al. 1994, pp. 8-9).
For example, let 
be 1 on the countable set 
and 0 elsewhere. At 0, this function
is approximately continuous but not continuous.
The exceptional set 
has Lebesgue measure 0
but accumulates at 0.
