College Math Teaching

January 14, 2019

New series in calculus: nuances and deeper explanations/examples

Filed under: calculus, cantor set — Tags: — collegemathteaching @ 3:07 am

Though I’ve been busy both learning and creating new mathematics (that is, teaching “new to me” courses and writing papers to submit for publication) I have not written much here. I’ve decided to write up some notes on, yes, calculus. These notes are NOT for the average student who is learning for the first time but rather for the busy TA or new instructor; it is just to get the juices flowing. Someday I might decide to write these notes up more formally and create something like “an instructor’s guide to calculus.”

I’ll pick topics that we often talk about and expand on them, giving suggested examples and proofs.

First example: Continuity. Of course, we say f is continuous at x = a if lim_{x \rightarrow a} f(x) = f(a) which means that the limit exists and is equal to the function evaluated at the point. In analysis notation: for all \epsilon > 0 there exists \delta > 0 such that |f(a)-f(x)| < \epsilon whenever |a-x| < \delta .

Of course, I see this as “for every open U containing f(a) , f^{-1}(U) is an open set. But never mind that for now.

So, what are some decent examples other than the usual “jump discontinuities” and “asymptotes” examples?

A function that is continuous at exactly one point: try f(x) = x for x rational and f(x) = x^2 for x irrational.

A function that oscillates infinitely often near a point but is continuous: f(x) = xsin(\frac{1}{x}) for x \neq 0 and zero at x = 0 .

A bounded unction with a non-jump discontinuity but is continuous for all x \neq 0 : f(x) = sin(\frac{1}{x}) for x \neq 0 and zero at x = 0 .

An unbounded function without an asymptote but is continuous for all x \neq 0 f(x) = \frac{1}{x} sin(\frac{1}{x}) for x \neq 0 and zero at x = 0 .

A nowhere continuous function: f(x) = 1 for x rational, and 0 for x irrational.

If you want an advanced example which blows the “a function is continuous if its graph can be drawn without lifting the pencil off of the paper, try the Cantor function. (this function is continuous on [0,1] , has derivative equal to zero almost everywhere, and yet increases from 0 to 1.

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