In this section we introduce the concepts of limit and continuous function.
Intuitive Definitions
Let f (x) be a function from a subset of the real numbers to the real numbers, and let x0 be a
real number. We say that f (x) has a limit at x = x0 if as x approaches x0, f (x) approaches
some number L. We call the number L the limit of the function f (x) at x0, and we write
 f (x) = L |
|
If
f is the function whose graph is drawn below, then
f (x) = 2 and
f (x) = 5. Note that a function need not be defined at a particular value of
x
(that is,
x need not belong to the domain) in order for a limit to exist there.
Figure 2.1: Limit Examples
It may be that a function has a limit at x = x0 only if x approaches x0 from one side
or the other. In fact, the function may have two different limits at x0, depending on
the side from which x approaches x0. Such limits are called one-sided limits. If f
has the limit L when x approaches x0 from the left, we say f has left-hand limit
L at x0 and write
 f (x) = L |
|
We make the similar definition for the right-hand limit. An example of a function with one-sided
limits is shown below. What is the left-hand limit of f at -2? The right-hand limit?
Figure 2.2: An Example of a One-Sided Limit
If a function f has a limit L at x0, and L = f (x0), then f is said to be
continuous at x0. Note that this presupposes that x0 is in the domain of f, which
is not necessary when discussing limits in general. A function that is continuous at every
point in its domain is said to be a continuous function.
All of the elementary functions are continuous (because they are continuous at the
x-values where they are defined.
Sometimes we want to talk about the limit of a function as x approaches infinity or
negative infinity (∞ or - ∞). This is essentially the same idea: approaching
∞ means that x is getting larger and larger; approaching - ∞ means smaller
and smaller.
Rigorous Definitions
We now make precise the intuitive definitions of limit and continuity given above. Let
f be a function from some subset of the real numbers to the real numbers and let x0
be a real number. Then the function f is said to have limit L at x0 if for all
ε > 0, there exists a δ > 0 such that 0 < | x - x0| < δ implies | f (x) - L| < ε. If this is the case, we write
| f (x) = L |
|
As above, if a function f has a limit L = f (x0) at x0, then f is said to be
continuous at x0. A function that is continuous at every point in its domain is said to
be a continous function.
As an example of a proof that uses this definition, we show that the linear function
f (x) = 3x is continuous at x0 = 1. Given ε > 0, we choose δ = ε/3.
Suppose | x - 1| < δ. Then | f (x) - f (1)| = | 3x - 3| = 3| x - 1| < 3δ = ε. Therefore the
limit of f (x) at x = 1 is f (1) = 3, and f is continuous there.
Intermediate Value Theorem
We conclude by mentioning an important property of continuous functions. Suppose
f (x) is continuous on an interval [a, b]. Let y be any number between f (a) and
f (b). Then the intermediate value theorem states that there exists c in the
interval (a, b) such that f (c) = y.
Roughly speaking, a continuous function cannot suddenly "jump" from one value to another (at the point
where such a jump would occur, the limit would not exist), so it must move from one value to another
by passing through every value in between. This is the way one should remember the idea behind the
intermediate value theorem.
