Step 1: Understanding the Concept:
A function is continuous at a point \(x=a\) if the limit as \(x \to a\) exists and is equal to the value of the function at that point.
If a function is not defined at a certain point, it is automatically discontinuous there.
The absolute value function \(|x|\) behaves differently for positive and negative numbers:
- For \(x \geq 0, |x| = x\)
- For \(x<0, |x| = -x\)
Step 2: Detailed Explanation:
Let's analyze the expression for \(f(x)\) based on the sign of \(x\).
Case 1: When \(x>0\).
Here, \(|x| = x\).
\[ f(x) = \frac{x - x}{x} = \frac{0}{x} = 0 \]
Case 2: When \(x<0\).
Here, \(|x| = -x\).
\[ f(x) = \frac{x - (-x)}{x} = \frac{x + x}{x} = \frac{2x}{x} = 2 \]
Case 3: When \(x = 0\).
The function involves division by \(x\). Since division by zero is undefined, \(f(0)\) does not exist.
Now let's check the limits as \(x\) approaches 0.
The Right-hand Limit (RHL) as \(x \to 0^+\):
\[ \lim_{x \to 0^+} f(x) = 0 \]
The Left-hand Limit (LHL) as \(x \to 0^-\):
\[ \lim_{x \to 0^-} f(x) = 2 \]
Since \(\text{LHL} \neq \text{RHL}\), the overall limit \(\lim_{x \to 0} f(x)\) does not exist.
Because there is no limit at \(x=0\) and the function is not defined at \(x=0\), the function has a jump discontinuity at the origin.
Evaluation of options:
(A) Incorrect - it is discontinuous at \(x=0\).
(B) Correct - it is a general statement that the function is not continuous.
(C) This is a true statement (it is constant \(2\) for all \(x<0\)), but it doesn't describe the function's overall continuity.
(D) This is also true, as it's continuous everywhere except for the single point at zero.
However, in standard testing, if a function is discontinuous even at one point, it is categorized as "not continuous".
Step 3: Final Answer:
The function is not continuous because it is undefined and the limit does not exist at \(x=0\).