Title: A Functional Bestiary
1A Functional Bestiary
Dan Kennedy Baylor School Chattanooga, TN
2From 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.
3Most 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!
4Polynomials
Trig Functions
Rational Functions
Exponential Functions
Log Functions
Logistic Functions
5Absolute Value
Algebraic Functions
6Greatest Integer
Piecewise-Defined
7Loosely speaking, a function is continuous at a
point if it can be graphed through that point
with an unbroken curve.
Continuous at x 0
Discontinuity at x 0
Removable Discontinuity at x 1
8Loosely 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.
9Points of non-differentiability include corners
(e.g., absolute value) points of verticality
(e.g., cube root) cusps (e.g., cube root of
absolute value)
10And, of course, points of discontinuity are also
points of non-differentiability.
11So 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.
12This 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.
13A true beast would fail even to have a right-hand
limit or a left-hand limit! Heres a good one
The right-hand and left-hand limits at 0 diverge
by oscillation.
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15In 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.
16How about a function with domain all real numbers
that is continuous nowhere? Heres one
This is called a salt-and-pepper function.
It cannot, of course, be drawn accurately.
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18The real definition of continuity A function f
is continuous at a if
Note, however, that the function
has no limit at any a. Now thats a beast!
19How 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
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21How about a function with domain all real numbers
that is continuous everywhere except at the
integers? Thats not too hard. The greatest
integer function is such a beast.
22We 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
231
2
3
1
2
3
24It 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.
25Extend f to a function of period 1 on the whole
real line.
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27You want proof? A careful proof involves
epsilons and deltas, but heres 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.
28For any irrational number I, the continuity
requirement boils down to
For irrational x, 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
29No matter how close the blue line is to zero,
only a finite number of dots are above it.
30So you can always find a small enough
neighborhood around I so that none of the points
are above it!
31So now let us consider the beasts of
non-differentiability. The interesting
ones are the continuous functions that fail to be
differentiable.
32Continuous but not differentiable corners (e.g.,
absolute value, ) points of verticality
(e.g., cube root) cusps (e.g., cube root of
absolute value)
33A warm-up Find a function that is continuous
for all real numbers but non-differentiable at
every integer.
34One of the strangest beasts in function history
is Karl Weierstrasss 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!
35Define G(x) to be the distance from x to the
nearest integer to x.
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37Note 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.
38Thanks to Tom Vogels Gallery of Calculus
Pathologies!
39Some 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.
40x y x y x y x y
0 41 10 151 20 461 30 971
1 43 11 173 21 503 31 1033
2 47 12 197 22 547 32 1097
3 53 13 223 23 593 33 1163
4 61 14 251 24 641 34 1231
5 71 15 281 25 691 35 1301
6 83 16 313 26 743 36 1373
7 97 17 347 27 797 37 1447
8 113 18 383 28 853 38 1523
9 131 19 421 29 911 39 1601
All these function values are primes!
41Heres a harmless-looking function that caused
trouble on the AB Calculus AP examination one
year
For one point, students had to give its domain.
42Sadly, some students simplified the function a
little too well
Nowis 0 still in the domain?
43George 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.
44We will define this beast on 0, 1 and then
extend it to a function on the whole real line.
We start with
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46The function we want is simply
It has range 0, 1. It is constant on intervals
of total measure
47Finally, 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!
48Well finish todays look at the Bestiary with a
pair of beasts that every young calculus student
should know. First, consider these two limits
49This function is continuous at x 0. It does
not have a derivative at x 0. It does not
exhibit local linearity at the origin.
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51This function has a derivative at x 0. You can
show it by using the definition of the derivative
at x 0. It also exhibits local linearity at
the origin.
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53Heres the derivative at x 0
54Heres the derivative for x ? 0
Note that this function does not have a limit as
x approaches 0. So f is differentiable
everywhere, but the derivative is not continuous
at 0.
55Perhaps more significantly, this function has a
derivative at 0, but you can not find it by
comparing the derivative on one side of 0 to the
derivative on the other side of 0. This is often
how students show that split functions are
differentiable. Luckily for them, that works for
nice functions. But not all the functions in the
FUNCTIONAL BESTIARY!
56dkennedy_at_baylorschool.org