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C. CONTINUITY AND
DISCONTINUITY
1. One-sided limits
We begin by expanding the notion of limit to include what are called one-sided limits, where x approaches a only from one side — the right or the left. The terminology and notation is:.
right-hand limit lim x→a+
f (x) (x comes from the right, x > a)
left-hand limit lim x→a−^
f (x) (x comes from the left, x < a)
Since we use limits informally, a few examples will be enough to indicate the usefulness of this idea.
-1^1
x
f(x)
f(x)
1 / x
1/x 2
Ex. 1 Ex.2 Ex. 3 Ex.
Example 1. lim x→ 1 −
1 − x^2 = 0 lim x→− 1 +
1 − x^2 = 0
(As the picture shows, at the two endpoints of the domain, we only have a one-sided limit.)
Example 2. Set f (x) =
− 1 , x < 0 1 , x > 0.
Then lim x→ 0 −
f (x) = − 1 , lim x→ 0 +
f (x) = 1.
Example 3. lim x→ 0 +
x
= ∞, lim x→ 0 −
x
Example 4. lim x→ 0 +
x^2
= ∞, lim x→ 0 −
x^2
The relationship between the one-sided limits and the usual (two-sided) limit is given by
(1) lim x→a f (x) = L ⇐⇒ lim x→a−
f (x) = L and lim x→a+
f (x) = L
In words, the (two-sided) limit exists if and only if both one-sided limits exist and are equal. This shows for example that in Examples 2 and 3 above, lim x→ 0 f (x) does not exist.
Students often say carelessly that lim x→ 0 1 /x = ∞, but this is not sloppy, it is simply
wrong, as the picture for Example 3 shows. By contrast, lim x→ 0 1 /x^2 = ∞ is correct and
acceptable terminology. 1
2
2. Continuity
To understand continuity, it helps to see how a function can fail to be continuous. All of the important functions used in calculus and analysis are continuous except at isolated points. Such points are called points of discontinuity. There are several types. Let’s begin by first recalling the definition of continuity (cf. book, p. 75).
(2) f (x) is continuous at a if lim x→a f (x) = f (a).
Thus, if a is a point of discontinuity, something about the limit statement in (2) must fail to be true.
Types of Discontinuity
sin (1/ x )
x x
2 1
removable removable jump infinite essential
In a removable discontinuity, lim x→a f (x) exists, but lim x→a f (x) 6 = f (a). This may be because
f (a) is undefined, or because f (a) has the “wrong” value. The discontinuity can be removed by changing the definition of f (x) at a so that its new value there is lim x→a f (x). In the left-most
picture, x^2 − 1 x − 1
is undefined when x = 1, but if the definition of the function is completed
by setting f (1) = 2, it becomes continuous — the hole in its graph is “filled in”.
In a jump discontinuity (Example 2), the right- and left-hand limits both exist, but are not equal. Thus, lim x→a f (x) does not exist, according to (1). The size of the jump is
the difference between the right- and left-hand limits (it is 2 in Example 2, for instance). Though jump discontinuities are not common in functions given by simple formulas, they occur frequently in engineering — for example, the square waves in electrical engineering, or the sudden discharge of a capacitor.
In an infinite discontinuity (Examples 3 and 4), the one-sided limits exist (perhaps as ∞ or −∞), and at least one of them is ±∞.
An essential discontinuity is one which isn’t of the three previous types — at least one of the one-sided limits doesn’t exist (not even as ±∞). Though sin(1/x) is a standard simple example of a function with an essential discontinuity at 0, in applications they arise rarely, presumably because Mother Nature has no use for them.
We say a function is continuous on an interval [a, b] if it is defined on that interval and continuous at every point of that interval. (At the endpoints, we only use the approrpiate one-sided limit in applying the definition (2).)
