for a wide class of functions "f" (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). the density "l" is (more exactly, can and will be chosen to be) continuous. the number "l"("x") is called the local time at "x" of "w" on [0, "t"]. it is strictly positive for all "x" of the interval ("a", "b") where "a" and "b" are the least and the greatest value of "w" on [0, "t"], respectively. (for "x" outside this interval the local time evidently vanishes.) treated as a function of two variables "x" and "t", the local time is still continuous. treated as a function of "t" (while "x" is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of "w".