![]() ![]() Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. In this sense the motivation to study continuity is not much different than the motivation to study, e.g., integers: Not every number occuring in real life is an integer but many are and properties of integers may thus be of general interest.Look up calculus in Wiktionary, the free dictionary. The above motivation translates to almost all of abstract mathematics for most people and most non-mathematicians: We detect and abstract some properties of natural objects and see whether we can derive some new abstract statements from those properties, which we can in turn translate back to application, where they may be useful. Moreover, only by knowing what we usually take as given due to the ubiquity of continuity, we know what we cannot expect anymore in the rare case that continuity is absent. Without having a concept of continuity, we would have to appeal to intuition, make assumptions or explicitly show the fact, whenever we would normally apply the intermediate-value theorem or swap limits and function evaluation. Moreover, it is worth studying whether any useful properties follow from continuity – which is indeed the case – as those properties would have a broad application. Therefore, it is only natural to have a word for this property and to properly define it to ensure that everybody is talking about the same thing. For example, we treat space and time as continuous, though they may be discretised on levels that we cannot yet access with measurements, and we can treat matter as continuous on many scales, if even though it is not continuos on the atomic level. ![]() How do you define $3^$" are unintuitive and often create confusion to high school/freshman college students.Ĭontinuity is a property that many real functions or relations have on practically relevant scales, i.e., if there are discontinuities, they are so small that they have no impact on most applications. It provides a natural way of understanding irrational exponents (or more generally, extend a function defined on rational number to real number). Whenever we think $f(x)$ can be approximated by $f(y)$ because $x$ can be approximated by $y$ we inherently assume $f$ is continuous. It's the foundation of most approximations we make in physical science. We use continuity very frequently without realizing we're using it. What is a good response to this question? Imagine a student asks the question why it is worth it to study continuity. They have to be shown the differences between continuity and the lack of it. This silly example shows me every year that my students do not understand how some functions can act differently from others. This assumption is correct, but when I ask them they cannot even imagine any discontinuous possibility such as a Star Trek beam transporter. Students assume that the distance from my home is a continuous function, so I must at some point be exactly ten miles from my home (school is 22 miles away). This student will get into trouble when this is not the case.Įxample story: when I begin to teach the Intermediate Value Theorem for continuous functions, I give the example of me driving to school from my home. ![]() Catastrophe theory, developed in the 1970's, shows there are a variety of ways to get discontinuity, with a variety of applications.Īny student who does not directly think about continuous and discontinuous functions early in his/her math training may assume that all functions are continuous.The study of Chaos, resulting from weather and other such topics, gives results that are practically discontinuous, even if continuous in theory. ![]() Schrodinger's equation assumes continuity and differentiability, but in the Copenhagen interpretation of quantum theory any measurement causes a collapse of the waveform and things become discontinuous: the spin is either up or down, etc. Quantum theory is a mixture of the continuous and the discontinuous.Many computer routines produce discontinuous output, even if the data is near-continuous. Computers and their digitization of data.However, more and more discontinuous functions are appearing in the various sciences. Most functions that are studied by physicists and other scientists are continuous. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |