Counterexamples

Abstract

The use of counterexamples to disprove theorems is discussed and contrasted with proof. Various counterexamples to plausible statements are presented. In particular, it is shown that matrices do not always commute, that matrix inverses do not always exist, and the square of a nonzero matrix can be zero. It is shown that cubing gives an infinitely differentiable bijection with non-differentiable inverse and zero derivative somewhere. A discontinuous function with the intermediate value property is presented as a counterexample to the idea that continuity can be defined via that property. A counterexample showing that limits of sequences do not commute is analyzed. An example of a first-order ordinary differential equation with multiple solutions is also presented. The use of determinants to characterize invertibility as an alternative way to establish the existence of non-invertible matrices is also discussed.