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

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 f whose derivative is discontinuous at one point x 0 and oscillates only above f ′ (x 0 ) (i.e. f ′ has a strict minimum at x 0 ), 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.