## Calculus AB and Calculus BC

## CHAPTER 2 Limits and Continuity

### 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*) = 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 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:

The graph of *f* is shown above.

We observe that *f* is not continuous at *x* = −2, *x* = 0, or *x* = 2.

At *x* = −2, *f* is not defined.

At *x* = 0, *f* is defined; in fact, *f* (0) = 2. However, since and does not exist. Where the left- and right-hand limits exist, but are different, the function has a *jump discontinuity.* The greatest-integer (or step) function, *y* = [*x*], has a jump discontinuity at every integer.

At *x* = 2, *f* is defined; in fact, *f* (2) = 0. Also, the limit exists. However, This discontinuity is called *removable.* If we were to redefine the function at *x* = 2 to be *f* (2) = −2, the new function would no longer have a discontinuity there. We cannot, however, “remove” a jump discontinuity by any redefinition whatsoever.

Whenever the graph of a function *f* (*x*) has the line *x* = *a* as a vertical asymptote, then *f* (*x*) becomes positively or negatively infinite as *x* → *a*^{+} or as *x* → *a*^{−}. The function is then said to have an *infinite discontinuity.* See, for example, Figure N2–4 for Figure N2–5 for or Figure N2–7 for Each of these functions exhibits an infinite discontinuity.

**EXAMPLE 24**

is not continuous at *x* = 0 or = −1, since the function is not defined for either of these numbers. Note also that neither nor exists.

**EXAMPLE 25**

Discuss the continuity of *f*, as graphed in Figure N2–9.

**SOLUTION:** *f* (*x*) is continuous on [(0,1), (1,3), and (3,5)]. The discontinuity at *x* = 1 is removable; the one at *x* = 3 is not. (Note that *f* is continuous from the right at *x* = 0 and from the left at *x* = 5.)

**FIGURE N2–9**

In Examples 26 through 31, we determine whether the functions are continuous at the points specified:

**EXAMPLE 26**

Is continuous at *x* = −1?

**SOLUTION:** Since *f* is a polynomial, it is continuous everywhere, including, of course, at *x* = −1.

**EXAMPLE 27**

Is continuous (a) at *x* = 3; (b) at *x* = 0?

**SOLUTION:** This function is continuous except where the denominator equals 0 (where *g* has an infinite discontinuity). It is not continuous at *x* = 3, but is continuous at *x* = 0.

**EXAMPLE 28**

Is continuous

(a) at *x* = 2; (b) at *x* = 3?

**SOLUTIONS:**

**(a)** *h*(*x*) has an infinite discontinuity at *x* = 2; this discontinuity is not removable.

**(b)** *h*(*x*) is continuous at *x* = 3 and at every other point different from 2. See Figure N2–10.

**FIGURE N2–10**

**EXAMPLE 29**

Is (*x* ≠ 2) continuous at *x* = 2?

**SOLUTION:** Note that *k*(*x*) = *x* + 2 for all *x* ≠ 2. The function is continuous everywhere except at *x* = 2, where *k* is not defined. The discontinuity at 2 is removable. If we redefine *f* (2) to equal 4, the new function will be continuous everywhere. See Figure N2–11.

**FIGURE N2–11**

**EXAMPLE 30**

Is continuous at *x* = 1?

**SOLUTION:** *f* (*x*) is not continuous at *x* = 1 since This function has a jump discontinuity at *x* = 1 (which cannot be removed). See Figure N2–12.

**FIGURE N2–12**

**EXAMPLE 31**

Is continuous at *x* = 2?

**SOLUTION:** *g*(*x*) is not continuous at *x* = 2 since This discontinuity can be removed by redefining *g*(2) to equal 4. See Figure N2–13.

**FIGURE N2–13**

THEOREMS ON CONTINUOUS FUNCTIONS

**(1) The Extreme Value Theorem.** If *f* is continuous on the closed interval [*a*,*b*], then *f* attains a minimum value and a maximum value somewhere in that interval.

**(2) The Intermediate Value Theorem.** If *f* is continuous on the closed interval [*a*,*b*], and *M* is a number such that *f* (*a*) ≤ *M* ≤ *f* (*b*), then there is at least one number, *c*, between *a* and *b* such that *f* (*c*) = *M*.

Note an important special case of the Intermediate Value Theorem:

If *f* is continuous on the closed interval [*a*,*b*], and *f* (*a*) and *f* (*b*) have opposite signs, then *f* has a zero in that interval (there is a value, *c*, in [*a*,*b*] where *f* (*c*) = 0).

**(3) The Continuous Functions Theorem.** If functions *f* and *g* are both continuous at *x* = *c*, then so are the following functions:

(a) *kf*, where *k* is a constant;

(b) *f* ± *g*;

(c) *f* · *g*;

(d) provided that *g*(*c*) ≠ 0.

**EXAMPLE 32**

Show that has a root between *x* = 2 and *x* = 3.

**SOLUTION:** The rational function *f* is discontinuous only at and *f* (3) = 1. Since *f* is continuous on the interval [2,3] and *f* (2) and *f* (3) have opposite signs, there is a value, *c*, in the interval where *f* (*c*) = 0, by the Intermediate Value Theorem.

### Chapter Summary

In this chapter, we have reviewed the concept of a limit. We’ve practiced finding limits using algebraic expressions, graphs, and the Squeeze (Sandwich) Theorem. We have used limits to find horizontal and vertical asymptotes and to assess the continuity of a function. We have reviewed removable, jump, and infinite discontinuities. We have also looked at the very important Extreme Value Theorem and Intermediate Value Theorem.