A Functional Bestiary

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A Functional Bestiary
Dan Kennedy Baylor School Chattanooga, TN
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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.
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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!
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Polynomials
Trig Functions
Rational Functions
Exponential Functions
Log Functions
Logistic Functions
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Absolute Value
Algebraic Functions
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Greatest Integer
Piecewise-Defined
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Loosely 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
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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.
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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)
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And, of course, points of discontinuity are also
points of non-differentiability.
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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.
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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.
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A 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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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.
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How 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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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 thats a beast!
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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
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How 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.
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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
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1
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1
2
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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.
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Extend f to a function of period 1 on the whole
real line.
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You 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.
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For 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
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No matter how close the blue line is to zero,
only a finite number of dots are above it.
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So you can always find a small enough
neighborhood around I so that none of the points
are above it!
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So now let us consider the beasts of
non-differentiability. The interesting
ones are the continuous functions that fail to be
differentiable.
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Continuous but not differentiable corners (e.g.,
absolute value, ) points of verticality
(e.g., cube root) cusps (e.g., cube root of
absolute value)
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A warm-up Find a function that is continuous
for all real numbers but non-differentiable at
every integer.
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One 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!
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Define G(x) to be the distance from x to the
nearest integer to x.
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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.
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Thanks to Tom Vogels Gallery of Calculus
Pathologies!
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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.
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All these function values are primes!
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Heres a harmless-looking function that caused
trouble on the AB Calculus AP examination one
year
For one point, students had to give its domain.
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Sadly, some students simplified the function a
little too well
Nowis 0 still in the domain?
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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.
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We 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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The function we want is simply
It has range 0, 1. It is constant on intervals
of total measure
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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!
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Well finish todays look at the Bestiary with a
pair of beasts that every young calculus student
should know. First, consider these two limits
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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.
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This 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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Heres the derivative at x 0
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Heres 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.
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Perhaps 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!
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dkennedy_at_baylorschool.org