A local level set of a Takagi-van der Waerden function is an equivalence class
under a digit-based relation finer than equality of function values. Fix an integer 
and write the base-
expansion of 
as
 |
with 
,
choosing the terminating expansion when there are two. Let 
be 1 when the fractional
part of 
is less than 
and 
otherwise, and put 
.
Two points 
and 
belong to the same local level set iff, for every 
, 
and, whenever 
, either 
or 
(Jiang et al. 2026). Each such
equivalence class is contained in a level
set of 
.
When 
,
the digit condition is automatic, giving the classical definition for the blancmange
function (Jiang et al. 2026).
