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Cell[CellGroupData[{
Cell["A note about continuity", "Subtitle"],
Cell["\<\
I. Cnop
Vrije Universiteit Brussel
icnop@vub.ac.be\
\>", "Text",
FontSlant->"Italic"],
Cell[CellGroupData[{
Cell["Considerations about topology", "Subsubsection"],
Cell["\<\
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.\
\>", \
"Text"]
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Cell[CellGroupData[{
Cell["Technology offers the incentive to engage in a new course. ", \
"Subsubsection"],
Cell["\<\
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:
\ta. no growth as seen with horizontal tangents (as in Rolle's theorem),
\tb. estimating growth by derivatives,
\tc. the order of contact between graphs;
- fitting data by well chosen functions, depending on the set of data:
\ta. polynomial approximation near a point,
\tb. Fourier approximations for nearly periodic data,
\tc. 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
\t(these preserve properties and all good approximations present some kind of \
uniformity somewhere).\
\>", "Text"]
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