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
ContinuousThe 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) = 0. 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