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Consider the following two statements about the function \(f(x)=|x|\): P. \(f(x)\) is continuous for all real values of \(x\). Q. \(f(x)\) is differentiable for all real values of \(x\). Which of the following is true?

1. P is true and Q is false.
2. P is false and Q is true.
3. Both P and Q are true.
4. Both P and Q are false.

Correct answer: P is true and Q is false.

Solution

The function \(|x|\) is continuous for every real \(x\) because there is no jump or break in its graph. However, it is not differentiable at \(x=0\) since the left-hand and right-hand derivatives are different there.

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