Question

The real-valued function \[ f(x) = \frac{|x - a|}{x - a} \] is analyzed as follows:

• A. \text{continuous only at} \( x = a \)
• B. \text{discontinuous only for} \( x > a \)
• C. \text{a constant function when} \( x > a \)
• D. \text{strictly increasing when} \( x < a \)
  \

Hint:

For piecewise-defined functions like \( \frac{|x-a|}{x-a} \), analyze different cases based on the modulus function and check continuity at critical points.

Answer 1

The Correct Option is C

Step 1: Analyze the function behavior
The given function is: \[ f(x) = \frac{|x - a|}{x - a}. \] Using the definition of the modulus function, we consider two cases: 1. Case 1: When \( x>a \) \[ |x - a| = x - a. \] Substituting in \( f(x) \): \[ f(x) = \frac{x - a}{x - a} = 1. \] Hence, for \( x>a \), \( f(x) \) is a constant function with value 1. 2. Case 2: When \( x<a \) \[ |x - a| = -(x - a) = a - x. \] Substituting in \( f(x) \): \[ f(x) = \frac{a - x}{x - a} = -1. \] Hence, for \( x<a \), \( f(x) \) is a constant function with value -1. Step 2: Check continuity at \( x = a \)
At \( x = a \), the function is undefined because the denominator becomes zero. This means \( f(x) \) is discontinuous at \( x = a \). Step 3: Conclusion
- The function is not continuous at \( x = a \). - The function is constant (equal to 1) for \( x>a \). - The function is also constant (equal to -1) for \( x<a \). - The function does not strictly increase or decrease for \( x<a \). Thus, the correct answer is: \[ \mathbf{\text{a constant function when } x>a.} \] \bigskip