Calculus: Analytic Geometry and Calculus, with Vectors

3.4 Continuity 149

is continuous. Prove the statement by filling in the intermediate steps in the
formula
lim P(x) P(a)
a-4a
and tell which theorems on limits are used in the process. Remark: The same
procedure shows that each polynomial is continuous.
2 Letting
Q(x) _

(x - 1) (x - 3)

(x - 2) (x - 4)'

show that Q is continuous at each x except 2 and 4.

3 Prove that the quotient of two functions is continuous wherever both

functions are continuous and the denominator is not zero. Remark: We recall

that the quotient of two polynomials is sometimes called a rational function.

Our results show that a rational function is continuous wherever the denominator

is not zero.

4 Determine the points of discontinuity of the functions f,, etcetera, for

which

(a) fi(x)= 1 + x2 (b) f2(x)^1

x

x2

(c) fa(x) = x(1^1 x) (d) f4(x) jxj

1 1

(e) fs(x) =x2 + 2x - 3

5 Does the assertion

(f) fn(x) =x2 + 2x + 3

approx f (x) =f(a)

..I=-al<s

abbreviate the assertion that to each positive number a there corresponds a posi-

tive number 3 such that lf(x) - f(a)i < e whenever Ix - al < 5? Ins.: It

can, but it does only if we agree that it does.

6 Taxi fare is 50 cents plus 10 cents for each quarter mile or fraction

thereof. Letting f(x) denote the fare for a ride of x miles, sketch a graph of f

and tell where f is discontinuous.

7 Assume (as is not quite true) that it takes 0.5 calorie of^220

heat to raise the temperature of 1 gram of ice 1 degree centigrade, u0

that it takes 80 calories to melt the ice at 0°C, and that it takes e0-

1.0 calorie to raise the temperature of I gram of water one degree so-

centigrade. Supposing that -40 S x < 20, let Q(x) be the +0-

number of calories of heat required to raise one gram of H2O

from temperature -40°C to x°C. Sketch a graph of Q. Ans.: ,O _20TITu

Figure 3.491.

8 The magnitude of the gravitational force which the earth Figure 3.491

exerts upon a particle is called its weight W. Suppose (as would

be true in the mechanics of Newton if the earth were a homogeneous spherical

ball) that there exist constants kl and k2 such that

W=klx (05x<R)

W=xs (xzR),