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A Functional Bestiary

A Functional Bestiary

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A Functional Bestiary

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  1. A Functional Bestiary Dan Kennedy Baylor School Chattanooga, TN

  2. From WikipediA, the free encyclopedia: A bestiary, or Bestiarum vocabulum is a compendium of beasts. Bestiaries were made popular in the Middle Ages in illustrated volumes that described various animals, birds and even rocks. The natural history and illustration of each beast was usually accompanied by a moral lesson.

  3. Most functions we encounter in the real world are fairly tame and domesticated. They are continuous. They are differentiable. They are smooth. Have a nice day!

  4. Exponential Functions Rational Functions Logistic Functions Trig Functions Log Functions Polynomials

  5. Algebraic Functions Absolute Value

  6. Piecewise-Defined Greatest Integer

  7. Loosely speaking, a function is continuous at a point if it can be graphed through that point with an unbroken curve. Removable Discontinuity at x = 1 Discontinuity at x = 0 Continuous at x = 0

  8. Loosely speaking, a function is differentiable at a point if its graph resembles a non-vertical line when you zoom in closely enough at that point.

  9. Points of non-differentiability include: corners (e.g., absolute value); points of verticality (e.g., cube root); cusps (e.g., cube root of absolute value)

  10. And, of course, points of discontinuity are also points of non-differentiability.

  11. So now let us look at a few discontinuous functions from our functional bestiary. Usually we see functions that are discontinuous at a single point. They make their point… but they are not very beastly.

  12. This function does not have a limit at 0, so it cannot be continuous there. It does have a right-hand limit and a left-hand limit, but the two are not equal. This is the wimpy way to foil continuity at a point.

  13. A true beast would fail even tohave a right-hand limit or a left-hand limit! Here’s a good one: The right-hand and left-hand limits at 0 diverge by oscillation.

  14. In the limit, this function pretty much smears itself against the entire interval [-1, 1] on the y-axis. Since a limit at 0 must be unique, the points cannot all be limits. So, we call them cluster points.

  15. How about a function with domain all real numbers that is continuous nowhere? Here’s one: This is called asalt-and-pepperfunction. It cannot, of course, be drawn accurately.

  16. The real definition of continuity… A function f is continuous at a if Note, however, that the function has no limit at any a. Now that’s a beast!

  17. How about a function with domain all real numbers that is continuous only at x = 0? You can construct one by making a slight alteration to the salt-and-pepper function

  18. How about a function with domain all real numbers that is continuous everywhere except at the integers? That’s not too hard. The greatest integer function is such a beast.

  19. We have a beast that is continuous everywhere except at the integers. How about a function that is discontinuous everywhere except at the integers? Pass the salt and pepper…

  20. –3 –2 –1 1 2 3

  21. It is now time to introduce my favorite beast of discontinuity: a function that is continuous at every irrational number… but discontinuous at every rational number! I am not making this up.

  22. Extend f to a function of period 1 on the whole real line.

  23. You want proof? A careful proof involves epsilons and deltas, but here’s the gist. The function is discontinuous at every rational for the same reason as the various salt-and-pepper functions. No matter how small 1/n is, it is too far away from 0 to share a 0 limit with its irrational neighbors.

  24. For any irrational number I, the continuity requirement boils down to For irrationalx, the function values are already at 0. For rational x, the key is that only a finite number of function values can be very far from zero. Consider the picture again…

  25. No matter how close the blue line is to zero, only a finite number of dots are above it.

  26. So you can always find a small enough neighborhood around I so that none of the points are above it!

  27. So now let us consider the beasts of non-differentiability. The interesting ones are the continuous functions that fail to be differentiable.

  28. Continuous but not differentiable: corners (e.g., absolute value, ); points of verticality (e.g., cube root); cusps (e.g., cube root of absolute value)

  29. A warm-up: Find a function that is continuous for all real numbers but non-differentiable at every integer.

  30. One of the strangest beasts in function history is Karl Weierstrass’s function that is continuous everywhere but differentiable nowhere! Since his time, simpler functions with this property have been constructed. Also, we now know that “most” functions that are continuous everywhere are, in fact, differentiable nowhere!

  31. Define G(x) to be the distance from x to the nearest integer to x.

  32. Note that each of these functions has half the amplitude and half the period of its predecessor. Now add them all up: This function is continuous everywhere and differentiable nowhere.

  33. Thanks to Tom Vogel’s Gallery of Calculus Pathologies!

  34. Some miscellaneous functions from the bestiary… This is a pretty normal-looking quadratic function until you carefully consider a table of values… …which we do in the next slide.

  35. All these function values are primes!

  36. Here’s a harmless-looking function that caused trouble on the AB Calculus AP examination one year: For one point, students had to give its domain.

  37. Sadly, some students simplified the function a little too well: Now…is 0 still in the domain?

  38. George Rosenstein showed me this one. How about a continuous function with domain all real numbers that has range all real numbers and a zero derivative almost everywhere? For comparison, the greatest integer function has a zero derivative almost everywhere… but, of course, it does not have range all real numbers.

  39. We will define this beast on [0, 1] and then extend it to a function on the whole real line. We start with

  40. The function we want is simply It has range [0, 1]. It is constant on intervals of total measure

  41. Finally, extend the function to make a continuous “ladder” that has domain all real numbers. The function will have range all real numbers, and its derivative will be constant except on a set of measure zero… …by George!

  42. We’ll finish today’s look at the Bestiary with a pair of beasts that every young calculus student should know. First, consider these two limits:

  43. This function is continuous at x = 0. It does not have a derivative at x = 0. It does not exhibit “local linearity” at the origin.