Some counterexamples on the behaviour of real-valued functions and their derivatives

TL;DR

This paper explores surprising phenomena in basic calculus involving oscillating functions, derivatives, and inverse functions, revealing counterintuitive behaviors and clarifying definitions and theorems.

Contribution

It presents new counterexamples and clarifies the behavior of derivatives and inverse functions under less restrictive conditions.

Findings

A differentiable function with a strict minimum need not be monotonic near the minimum.

A derivative can be discontinuous at a point with a strict minimum, oscillating only in one direction.

Different definitions of inflection points are compared and analyzed.

Abstract

We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict minimum doesn't have to be decreasing to the left nor increasing to the right of the minimum, we present a function whose derivative is discontinuous at one point and has a strict minimum at this point (i.e. it oscillates only in one direction), we compare several definitions of inflection point, and we discuss a general version of the theorem on the derivative of inverse functions where continuity of the inverse function is assumed merely at one point.