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Question

Consider the function f(x)=sgn(x1) and g(x)=cot1[x1], where [.] denotes the greatest integer function.

Statement 1: The function F(x)=f(x)g(x) is discontinuous at x=1.
Statement 2: If f(x) is discontinuous at x=a and g(x) is also discontinuous at x=a, then the product function f(x)g(x) is discontinuous at x=a.

A
Both the statements are true and Statement 2 is the correct explanation of Statement 1.
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B
Both the statements are true and Statement 2 is not the correct explanation of Statement 1.
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C
Statement 1 is true and Statement 2 is false.
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D
Statement 1 is false and Statement 2 is true.
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Solution

The correct option is C Statement 1 is true and Statement 2 is false.
f(x)=sgn(x1);g(x)=cot1[x1]F(x)=f(x)g(x)=cot1[x1], x10, x=1cot1[x1], x1+=⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪3π4, x10, x=1π2, x1+F(1)=3π4;F(1+)=π2 and F(1)=0
F(x) is discontinuous at x=1
So, statement 1 is true but product of two discontinuous functions may be a continuous function.
So, statement 2 is false.

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