Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to:
- Computers and their digitization of data. Many computer routines produce discontinuous output, even if the data is near-continuous.
- Quantum theory is a mixture of the continuous and the discontinuous. 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.
- The study of Chaos, resulting from weather and other such topics, gives results that are practically discontinuous, even if continuous in theory.
- Catastrophe theory, developed in the 1970's, shows there are a variety of ways to get discontinuity, with a variety of applications.
Any student who does not directly think about continuous and discontinuous functions early in his/her math training may assume that all functions are continuous. This student will get into trouble when this is not the case.
Example 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. 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 assumption is correct, but when I ask them they cannot even imagine any discontinuous possibility such as a Star Trek beam transporter. This silly example shows me every year that my students do not understand how some functions can act differently from others. They have to be shown the differences between continuity and the lack of it.