Lower and upper level sets

Pixels with the gray values below threshold T belong to this lower level set.

Suppose we have a real valued function f.

Given a number (level, threshold, etc) T, 
the T-th lower/upper level set is the set of all inputs 
with the value of f lower/higher than or equal to T.

Or, these two sets for each T:

{x: f(x) ≥ T} and {x: f(x) ≤ T}.

A relevant result here is the Implicit Function Theorem.

They are also called "excursion sets" (why?).

Via thresholding we use these sets in analysis of gray scale images.