Nowhere monotone functions and functions of nonmonotonic type

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.