294 Functions, graphs, and numbers
has a tangent at each point where it does not have a corner. We can wonder
whether we can associate angles with corners in such a way that, whenever we
take a finite set of corners, the sum of the angles at these corners must be less
than or equal to 27r. We can wonder whether it is possible to construct a set S
such that I' has a corner at each of its points. We can ask questions much
faster than we can answer them. We can conclude that we have some very
substantial and useful information about tangents, but we do not yet know
everything.
5.2 Trends, maxima, and minima
means to say "it has been getting hotter all morning" or "the temperature
has been increasing all morning" should easily comprehend the following
definition. In most of the applications we shall meet, the set S will be a
set in El (that is, a set of real numbers) which is either (i) the whole set
of real numbers or (ii) the set of numbers in a closed interval a < x 5 b
or (iii) the set of numbers in an open interval a < x < b. However, S
can be the set of positive integers or any other set in which we may be
interested.
Definition 5.21 -1 function f is said to be increasing over a set S in El
if f (xl) < f (X2) whenever xl and x2 are two numbers in S for which xl < x2.
The function is said to be decreasing over S if f(xi) > f(x2) whenever xl and
x2 are two numbers in S for which xl < x2.
The following definition is more subtle. If the temperature was 30°
from 10:00 A.M. to 11:00 A.M., an articulate and truthful person would not
be expected to say that the temperature was increasing from 10:00 A.M.
to 11:00 A.M. However, in accordance with the following definition, the
temperature might have been monotone increasing all morning.
Definition 5.22 -1 function f is said to be monotone increasing over a
set S in El if f (xl) 5 f (X2) whenever xl and x2 are two numbers in S for
which xl < X2. The function is said to be monotone decreasing over S if
f (XI) > f (X2) whenever xl and x2 are two numbers in S for which xl < x2.
The terminology in this definition is very useful, and it may seem to be
less than utterly foolish when we realize that f is called monotone (some
people have preferred the word monotonous) if it is either monotone
increasing or monotone decreasing. For
example, the function f having the graph(^1) 1 \ shown in Figure 5.23 is increasing over
A u the interval a < x < xi, is decreasing
xl x2 ° over theinterval x2 (^5) x< b, is mono-
Figure 5.23 tone increasing over the interval
a 5 x < X2, and is monotone decreasing
over the interval xl S x 5 b. To appreciate the necessity for the fol-
lowing definitions, it may be sufficient to realize that it is impossible to
be quite sure what is meant when someone says that "the temperature at
Pike's Peak reached a maximum at noon last Friday."