Step 1: Understanding the Concept:
Continuity of a function at a point requires that the limit from the left, the limit from the right, and the function's value at that point are all equal.
The function \(f(x)\) involves the absolute value function \(|x|\), which behaves differently for positive and negative values.
Additionally, the function has \(x\) in the denominator, which means the function is undefined at \(x = 0\).
Step 2: Detailed Explanation:
Recall the definition of \(|x|\):
\(|x| = x\) for \(x \ge 0\)
\(|x| = -x\) for \(x<0\)
Now, analyze \(f(x)\) for two cases:
**Case 1: \(x>0\)**
Substitute \(|x| = x\):
\[ f(x) = \frac{x - x}{x} = \frac{0}{x} = 0 \]
So, for all positive \(x\), the function is a constant 0. A constant function is continuous on its domain.
**Case 2: \(x<0\)**
Substitute \(|x| = -x\):
\[ f(x) = \frac{x - (-x)}{x} = \frac{2x}{x} = 2 \]
So, for all negative \(x\), the function is a constant 2. This is also continuous on its domain.
**Case 3: \(x = 0\)**
The function is undefined at \(x = 0\) due to division by zero.
Also, observe the limits:
Left-hand limit \(\lim_{x \to 0^-} f(x) = 2\).
Right-hand limit \(\lim_{x \to 0^+} f(x) = 0\).
Since the LHL \(\ne\) RHL, the limit does not exist, and there is a jump discontinuity at \(x = 0\).
Conclusion: The function is perfectly continuous everywhere it is defined (for all \(x \ne 0\)).
Step 3: Final Answer:
The function is continuous for all \(x\) except zero.
Hence, the correct option is (D).