   Chapter 2.5, Problem 39E

Chapter
Section
Textbook Problem

Show that f is continuous on ( –∞, ∞). f ( x ) = { 1 − x 2   if   x ≤ 1 ln x       if   x > 1

To determine

To show: The function f(x)={1x2if x1lnxif x>1 is continuous on (,).

Explanation

Definition used: “A function f is continuous at a number a if limxaf(x)=f(a)”.

Theorem used:

The functions such as “Polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, exponential functions and logarithmic functions” are continuous at every number in their domains.

Proof:

Consider the function f(x)={1x2if x1lnxif x>1.

Here, f(x)=1x2 is defined on the interval (,1). Since f(x)=1x2 is a polynomial and by Theorem, it is continuous on its domain (,1).

Also f(x)=lnx is defined on the interval (1,). Since f(x)=lnx is logarithmic function and by Theorem, it is continuous on its domain (1,).

Therefore, f(x) is continuous on (,1)(1,).

To show f(x) is continuous on (,), it is enough to show f(x) is continuous at x=1.

At x=1, f(x)=1x2. Substitute x=1 in f(x)=1x2 as follows,

f(1)=1(1)2=11=0

Therefore, f(1)=0 is defined

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