文档介绍：Section
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Limits And The Notion
 ¢¡£¡¥¤§¦¨¤§©  £
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Of particular interest in mathematics and its applications to the physical world are func-
tional relationships in which the dependent variable changes continuously with changes in
the independent variable. Intuitively, changing continuously means that small changes in
the independent variable do not produce abrupt changes in the dependent variable. For
example, a small change in the radius of a circle does not produce an abrupt change in
the area of the circle; we would say that the area of the circle changes continuously with
the radius of the circle. Similarly, a small change in the height from which some object
is dropped will result in a related small change in the object’s terminal velocity; hence
terminal velocity is a continuous function of height. On the other hand, when an electri-
cal switch is closed, there is an abrupt change in the current flowing through the circuit;
the current flow through the circuit is not a continuous function of time. The purpose
of this section is to introduce the terminology and concepts that will give us a proper
mathematical basis for discussing continuity in the next section.
To begin our study of continuity, we will first look at two examples of functions which
are not continuous. In this way we will discover what properties to exclude when forming
our definition of a continuous function.
Example Consider the function H defined by
0, if t < 0,
H(t) = ()
1, if t ≥ 0.
This function, known as the Heaviside function, might be used in connection with modeling
the current passing through a switch which is open until time t = 0 and then closed. The
graph of this function consists of two horizontal half-lines with a vertical gap of unit length
at the origin, as shown in Figure . Since this function has a break in its graph at 0,
its output changes abruptly as