   Chapter 2.5, Problem 101E

Chapter
Section
Textbook Problem

a. Prove that a polynomial function y = P(x) is continuous at every number x. Follow these steps: (i) Use Properties 2 and 3 of continuous functions to establish that the function g(x) = xn, where n is a positive integer, is continuous everywhere. (ii) Use Properties 1 and 5 to show that f(x) = cxn, where c is a constant and n is a positive integer, is continuous everywhere. (iii) Use Property 4 to complete the proof of the result. b. Prove that a rational function R(x) = p(x)/q(x) is continuous at every point x where q(x) ≠ 0. Hint: Use the result of part (a) and Property 6.

(a)

To determine

To prove: A polynomial function y=P(x) is continuous at every number x .

Explanation

Proof:

Property 1: The constant function f(x)=c is continuous everywhere.

Property 2: The identity function f(x)=x is continuous everywhere.

Property 3: If the function f(x) and g(x) is continuous at x=a , then [f(x)]n will be continuous at x=a .

Property 4: The function f±g is continuous at x=a when the function f(x) and g(x) is continuous at x=a .

Property 5: The function fg is continuous at x=a when the function f(x) and g(x) is continuous at x=a .

Property 6: The rational function f/g will be continuous at x=a provided that g(a)=0 when f and g is continuous at x=a .

The given polynomial function is,

y=P(x) (1)

Assume the function P(x)=anxn+an1xn1++a1x1+a0 .

The identity function f(x)=x is continuous everywhere by property (2).

If the function f(x) and g(x) is continuous at x=a , then [f(x)]n will be continuous at x=a by property 3.

From the property (2) and (3), f(x),g(x) and [f(x)]n is continuous at x=a

(b)

To determine

To prove: The rational function R(x)=p(x)/q(x) is continuous at every point x where q(x)0 .

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