A note about continuity

I. Cnop
Vrije Universiteit Brussel
icnop@vub.ac.be

Considerations about topology

have been over-emphasised in the standard mathematics curriculum. This has led to a pure mathematics curriculum that was mainly concerned with pathology, and that has been difficult to adapt to real-world situations or applied curricula. It has been damaging the reputation of mathematics as a subject, and a setback to education  in general in many countries.

Technology offers the incentive to engage in a new course.

Let us collect what is important in calculus:
- visual experiments that motivate students;
- the ability to make computations with arbitrary precision, e.g. exploit continuity at a point to locate zeroes;
- the growth of functions:
	a. no growth as seen with horizontal tangents (as in Rolle's theorem),
	b. estimating growth by derivatives,
	c. the order of contact between graphs;
- fitting data by well chosen functions, depending on the set of data:
	a. polynomial approximation near a point,
	b. Fourier approximations for nearly periodic data,
	c. bell shapes or wavelets for hump-like data;
- the speed of approximations and fastness of algorithms in general;
- modeling of real world problems and discussing the quality of the model;
- trying to do analysis on data for which the prescribed function is not yet known;
- the ever-present interplay between continuous and discrete mathematics;
- uniform continuity on closed intervals and uniform limits on closed intervals
	(these preserve properties and all good approximations present some kind of uniformity somewhere).