(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 3631766, 155747]*) (*NotebookOutlinePosition[ 3632430, 155770]*) (* CellTagsIndexPosition[ 3632386, 155766]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Uniform continuity of objects", "Subtitle"], Cell["from the 1997 Samos Conference", "Subsubtitle"], Cell["\<\ I. Cnop, Vrije Universiteit Brussels icnop@vub.ac.be\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ We start by illustrating the concept of a function by some \ measurement: temperature plotted over some time interval. Start by reading off temperatures from and old fashioned thermometer. This \ consists of a thin glass tube in which some liquid may expand, and the level \ is read off sideways onto some scale next to the tube. Mathematics, as \ opposed to other sciences does this reading with arbitrary precision, and at \ all time instants (also determined with arbitrary precision). In a museum one \ can still see mechanic devices for automatic plotting on a paper drum. But \ this is not essential here: we suppose the final result is some line graph \ over some time interval and we do not really care how it was obtained. If the readings are done with a thermometer of which the scale had been moved \ up or down, then this results into a vertical shift in the graph. Keep in \ mind that if the scale slides downwards, all readings will increase by the \ same amount. If the scale of the thermometer is sitting somewhat loose, it \ can be moved slightly off its central position and readings will be done \ within certain bounds (which can still be determined with arbitrary \ precision).This results in a band which replaces the original line graph. If the readings are plotted against a watch which was not set right (or with \ reference to a different time zone), then this results in a horizontal shift \ in the graph. Keep in mind that if the watch is running behind, the plot of \ the graph will shift to the right. We expect that readings differ little if readings are only slightly off \ correct time. This amounts to saying that there are no sudden jumps in \ temperature. -- this phenomenon will be called continuity It is moreover expected that such differences in readings over a small time \ interval depend little on the time of the actual reading -- this phenomenon will be called uniform continuity Wrapping this up we get the following definition of uniform continuity: If the graph of a function is shifted slightly up and down, it can be shifted \ over a sufficiently small distance sideways in the band between these \ vertically modified positions Traditionally this is called the epsilon-delta condition in which the \ vertical shift is some epsilon and the horizontal shift is given by some \ delta which has to be sufficiently small but positive. 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