Assertion (A) : `|sinx|` is continuous for all `x inR`.
Reason (R ) : `sinxand|x|` are continuous in R.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the question regarding the continuity of the function \( | \sin x | \) and the reasoning provided, we will analyze the assertion and the reason step by step.

### Step 1: Understanding the Assertion
The assertion states that \( | \sin x | \) is continuous for all \( x \in \mathbb{R} \).

**Hint:** Recall the definition of continuity at a point and consider the properties of the sine function.

### Step 2: Analyzing the Function \( \sin x \)
The sine function, \( \sin x \), is a well-known trigonometric function that is continuous for all \( x \in \mathbb{R} \). This means that there are no breaks, jumps, or holes in the graph of \( \sin x \).

**Hint:** Use the fact that trigonometric functions are continuous everywhere.

### Step 3: Understanding the Modulus Function
The modulus function \( |x| \) is also continuous for all \( x \in \mathbb{R} \). The modulus function takes any real number and converts it to its non-negative counterpart.

**Hint:** Consider how the modulus function behaves with negative and positive inputs.

### Step 4: Composition of Continuous Functions
The composition of two continuous functions is also continuous. Since \( \sin x \) is continuous and \( |x| \) is continuous, we can conclude that \( | \sin x | \) (which is \( |g(f(x))| \) where \( f(x) = \sin x \) and \( g(x) = |x| \)) is continuous.

**Hint:** Remember the theorem that states the composition of continuous functions is continuous.

### Step 5: Conclusion on the Assertion
Since both \( \sin x \) and \( |x| \) are continuous, it follows that \( | \sin x | \) is continuous for all \( x \in \mathbb{R} \). Therefore, the assertion is true.

### Step 6: Understanding the Reason
The reason states that both \( \sin x \) and \( |x| \) are continuous in \( \mathbb{R} \). This is indeed true and supports the assertion.

**Hint:** Check if the reason logically supports the assertion.

### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion. Thus, the answer to the question is:

- Assertion (A) is true.
- Reason (R) is true and is the correct explanation for (A).