We investigate the relationships between the notions of a continuous functi
on being monotone on no interval, monotone at no point, of monotonic type o
n no interval, and of monotonic type at no point. In particular, we charact
erize the set of all points at which a function that has one of the weaker
properties fails to have one of the stronger properties. A theorem of Garg
about level sets of continuous, nowhere monotone functions is strengthened
by placing control on the location in the domain where the level sets are l
arge. It is shown that every continuous function that is of monotonic type
on no interval has large intersection with every function in some second ca
tegory set in each of the spaces P-n, C-n, and Lip.