Oscillating Functions That Disprove Misconceptions on Real-Valued Functions

When preparing a lecture on basic calculus you are sometimes reminded of Hamlet’s famous words “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” In this spirit, the purpose of this paper is to draw the reader’s attention to some surprising observations on differentiable functions of one variable which have arisen from our own experience as lecturers. Some of them seem to be little-known or even unknown, others can already be found in at least one of the many excellent calculus textbooks, but are included since we consider them to be particularly instructive. We hope that this note is of interest to faculty who want to flavor their lectures with some unexpected or even weird examples and to students who want to learn about such examples. In this context we also refer the reader to the books of Appell [2], Gelbaum and Olmsted [7], and Rajwade and Bhandari [10] which contain rich collections of counterexamples and peculiarities from real analysis. The unifying theme in our counterexamples is the use of “oscillating” functions like x → x sin 1 xm (where k ≥ 0, m ≥ 1) and related functions. They are a popular source of enlightening counterexamples in basic calculus, see, for example, [7, Chapter 3]) and for a more detailed discussion of the properties of those functions [2, Section 4.5], [3, 4]. For instance, the function x → x2 sin 1 x2 shows that derivatives are not necessarily continuous, that they can even be unbounded on compact intervals, and that differentiable functions need not be rectifiable on compact intervals, i.e., their graphs can have infinite length. From the point of view of complex analysis, the interesting properties of these functions are of course related to the fact that z → z sin 1 zm has an essential singularity at z = 0. We present some further applications of functions of this kind that disprove popular misconceptions on the behavior of differentiable real functions. Some of those misconceptions seem to arise from the very same origin: Many students tend to think of differentiable functions as being piecewise monotonic and are unaware of the fact that the monotonicity behavior might change infinitely many times within a compact interval. Naturally, oscillating functions like x → sin 1 x are apt to show those students that their imagination might mislead them. This central idea will appear in several of our examples, either obviously (Examples 2 and 3) or in a somewhat “disguised” form (Example 5), in the sense that not the function but its derivative has accumulating changes of monotonicity (and hence accumulating critical points).