Chapter 5
Continuous Functions
Sections 5.1- 5.3 are among the most important in the entire
book. The ideas discussed here, especially in Section 5.3, are
quite powerful. Section 5.4 through Theorem 5.4.7, is also
important, but in a one-semester course can be delayed until
it is needed in Chapter 7. Section 5.5 is optional, but it does
contain a complete treatment of the Cantor function. Cover
Section 5.6 only if you want a rigorous "early" treatment
of exponential and logarithm functions. Section 5. 7 requires
advanced mathematical maturity.The concept of continuous functions is introduced in a typical freshman calculus
course, but is not investigated in depth there because applied concepts are
regarded as more important in that course. However, the concept of continuity
is of great importance in analysis. Briefly, you will recall the intuitive notion of
continuity: A function is "continuous" if you can draw its graph without lifting
your pencil from the paper. This definition is obviously too vague for rigorous
mathematical purposes. In beginning calculus, an attempt is made at being
more rigorous: there, a function is defined to be continuous at a point x 0 if
three conditions hold:
(a) f(xo) exists;(b) lim f(x) exists; and
X--+Xo
(c) lim f(x) = f(xo).
x--+xoThe definition we are going to use is equivalent to these three conditions
only when x 0 is a cluster point of the domain off. It is stated as an c-8 criterion.
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