276 Chapter 5 • Continuous Tunctions
(c) Let r E Q. Prove that if r > 0, the function f(x) = xr is positive,
continuous, and strictly increasing on (0, +oo); also if r < 0, the
function f(x) = xr is positive, continuous, and strictly decreasing
on (0, +oo).5.6 *Exponentials, Powers, and Logarithms
If "early" definition and use of exponential and logarithm
functions are not of concern in your course, this section should
be omitted. These functions will be reintroduced in Chapter
7.
This section is another demonstration of the power of the
monotone convergence theorem. It can be covered as a class
project, assigning portions of the material to small groups.
The proofs may seem a bit tedious.The trend in recent years has been toward introducing exponential and logarith-
mic functions earlier in elementary calculus courses than had been customary in
preceding decades. The motivation for this comes from wanting to make these
highly useful functions available for use in examples and applications as soon
as possible. This trend has resulted in a slightly embarrassing situation: these
functions are introduced and used before they have been rigorously defined.
Students are asked to believe claims about existence and continuity of these
functions, and related limits, on the basis of plausibility arguments. While ac-
ceptable in an elementary calculus course, such an argument does not meet the
standards of rigor required by a real analysis course.
This section is included here to make rigorous the notions of exponential,
power, and logarithm functions, in recognition of the trend to introduce these
functions as early as feasible.
The traditional approach taken in elementary calculus courses has been
to first define the natural logarithm, lnx = J 1 x tdt. (Indeed, we shall take
this approach in Section 7.7.) While this definition seems contrived and bears
little resemblance to the approach to logarithms taken in elementary algebra
courses, the function so defined is shown to have the usual algebraic properties
associated with logarithms. Strangely, these properties are derived using the
derivative of this function and properties of antiderivatives (strange, since these
properties seem to have nothing to do with derivatives). Since lnx is a strictly
increasing function, it has an inverse, which is denoted ex. This, too, is shown
to have the algebraic properties associated with exponential functions. When
this approach is taken, the derivative of ln x is obvious from its definition, while
the derivative of ex is found as the derivative of an inverse function. Finally,
the general exponential function ax (for a > 0) is defined by ax = exlna and
the general logarithm function is defined as the inverse of ax. It is easy to