Question

Let the function \[ f(x)=\frac{x-|x|}{x} \] Then

• A. the function is continuous everywhere
• B. the function is not continuous
• C. the function is continuous when \(x<0\)
• D. the function is continuous for all \(x\) except zero

Hint:

Always split modulus functions into cases: \[ x>0 \quad \text{and} \quad x<0 \] before checking continuity.

Answer 1

The Correct Option is D

Concept:
Use the definition of modulus function: \[ |x|= \begin{cases} x, & x\ge0 -x, & x<0 \end{cases} \]

Step 1: Evaluate the function for \(x>0\). For \(x>0\), \[ |x|=x \] Hence, \[ f(x)=\frac{x-x}{x} \] \[ f(x)=0 \] So the function is continuous for all \(x>0\).

Step 2: Evaluate the function for \(x<0\). For \(x<0\), \[ |x|=-x \] Therefore, \[ f(x)=\frac{x-(-x)}{x} \] \[ f(x)=\frac{2x}{x} \] \[ f(x)=2 \] So the function is continuous for all \(x<0\).

Step 3: Check continuity at \(x=0\). At \(x=0\), \[ f(x)=\frac{x-|x|}{x} \] is not defined because denominator becomes zero. Hence the function is not continuous at \(x=0\).

Step 4: Conclusion. The function is continuous everywhere except at \(x=0\). \[ \boxed{\text{Function is continuous for all }x\neq0} \] Hence the correct answer is: \[ \boxed{(D)} \]