Functions, Limits, and Continuity
Limits and Continuity
In this section we introduce the concepts of limit and continuous function.
Let f (x) be a function from a subset of the real numbers to the real numbers, and let x 0 be a real number. We say that f (x) has a limit at x = x 0 if as x approaches x 0 , f (x) approaches some number L . We call the number L the limit of the function f (x) at x 0 , 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.
It may be that a function has a limit at x = x 0 only if x approaches x 0 from one side or the other. In fact, the function may have two different limits at x 0 , depending on the side from which x approaches x 0 . Such limits are called one-sided limits. If f has the limit L when x approaches x 0 from the left, we say f has left-hand limit L at x 0 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?
If a function f has a limit L at x 0 , and L = f (x 0) , then f is said to be continuous at x 0 . Note that this presupposes that x 0 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.
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 x 0 be a real number. Then the function f is said to have limit L at x 0 if for all ε > 0 , there exists a δ > 0 such that 0 < | x - x 0| < 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 (x 0) at x 0 , then f is said to be continuous at x 0 . 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 x 0 = 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.