Abstract:

The aim of this article is to show that graphical software cannot fully explain the textbook definition of a function. The explanation is especially targeted at prospective mathematics teachers, who teach the proper definition of function to their students.

Graphical representation of functions is a tool which assists in the development of the concept of a function. Curriculum includes visualization of functions via their graphs. Computer assist helps, especially if the function is defined by an explicit equation of the form y=f(x). For instance, computers assist visualization of the function f(x)=(x+1)/(x4+1) by providing a graphic without the pointwise process of pencil and paper plotting.

Graphs produced by computer software are limited to a minor but important subset of the totality of all graphs possible to define by functions. The article discusses functions whose graphs are not produced by computer software. One example looks like the Cartesian plane painted black, instead of a curve. The main examples are additive functions, which are defined without an explicit equation y=f(x). An historical account of additive functions is given, together with ideas for integrating such pathological examples into teaching.

Keywords:

Cauchy’s functional equation, mathematics teaching.