Step 1: Understanding the Concept:
Continuity at a point \( x = a \) requires that the function is defined at \( a \), the limit exists at \( a \), and the limit equals the function's value.
A rational function is discontinuous where its denominator is zero.
Detailed Explanation:
Let's analyze \( f(x) = \frac{x-|x|}{x} \) by breaking it into pieces based on the definition of absolute value:
\( |x| = x \) for \( x>0 \) and \( |x| = -x \) for \( x<0 \).
1. For \( x>0 \):
\[ f(x) = \frac{x - x}{x} = \frac{0}{x} = 0 \]
The function is a constant 0 for all positive \( x \). Constant functions are continuous on their domains.
2. For \( x<0 \):
\[ f(x) = \frac{x - (-x)}{x} = \frac{2x}{x} = 2 \]
The function is a constant 2 for all negative \( x \). This is also continuous on its domain.
3. At \( x = 0 \):
The denominator is 0, making the function undefined. Therefore, \( f(0) \) does not exist.
Also, if we check the limits:
Right-hand limit (\( x \to 0^+ \)) is 0.
Left-hand limit (\( x \to 0^- \)) is 2.
Since the limits from both sides are not equal, the limit at 0 does not exist (jump discontinuity).
Since the function is perfectly continuous in the open intervals \( (-\infty, 0) \) and \( (0, \infty) \), the only point of discontinuity is at 0.
Step 3: Final Answer:
The function is continuous for all values of \( x \) except zero.