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Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640

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Section
BuyFindarrow_forward

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

In Exercises 85-97, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.

97. If f is discontinuous at a, and g is discontinuous at f(a), then gf must be discontinuous at a.

To determine
Whether the composite function gf is discontinuous at a if the function f is continuous at a and the function g is discontinuous at f(a) .

Explanation

The given statement is,

If f is discontinuous at a and g is discontinuous at f(a) , then gf must be discontinuous at a ,

Assume the function f(x) and g(x) is,

f(x)={1ifx02x+3ifx>0 (1)

g(x)={1ifx0x+5ifx>0 (2)

Thus the function g(x)f(x) is,

g(x)f(x)={1ifx0(2x+8)ifx>0 (3)

Apply left hand limit in equation(1),

limx0f(x)=limx01=1

Apply right hand limit in equation(1),

limx0+f(x)=limx0+2x+3=20+3=3

The value of the function at x=0 is,

f(0)=1

The function f(x) is discontinuous at x=0

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