The value of e can be approximated on a graphing calculator to a large number of decimal places by
evaluating
for large values of x.
F. CONTINUITY
If a function is continuous over an interval, we can draw its graph without lifting pencil from paper.
The graph has no holes, breaks, or jumps on the interval.
Conceptually, if f (x) is continuous at a point x = c, then the closer x is to c, the closer f (x) gets to
f (c). This is made precise by the following definition:
DEFINITION
The function y = f (x) is continuous at x = c if
(1) f (c) exists; (that is, c is in the domain of f );
(2) exists;
(3)
A function is continuous over the closed interval [a, b] if it is continuous at each x such that a ≤ x
≤ b.
A function that is not continuous at x = c is said to be discontinuous at that point. We then call x =
c a point of discontinuity.
CONTINUOUS FUNCTIONS
Polynomials are continuous everywhere; namely, at every real number.
Rational functions, are continuous at each point in their domain; that is, except where Q(x) =
- The function for example, is continuous except at x = 0, where f is not defined.
The absolute value function f (x) = |x| (sketched in Figure N2–3) is continuous everywhere.
The trigonometric, inverse trigonometric, exponential, and logarithmic functions are continuous at
each point in their domains.
Functions of the type (where n is a positive integer ≥ 2) are continuous at each x for which
is defined.
The greatest-integer function f (x) = [x] (Figure N2–1) is discontinuous at each integer, since it
does not have a limit at any integer.
KINDS OF DISCONTINUITIES
In Example 2, y = f (x) is defined as follows: