Let f(x)=|x|+[x-1], where [ . ] is greatest integer function , then f(x) is

Options

A. continuous at x=0 as well as at x=1
B. continuous at x=0 but not at x=1
C. continuous at x=1 but not at x=0
D. neither continuous at x=0 or nor at x=1

Answer

To analyze the function \( f(x) = |x| + [x-1] \), where \( [ \cdot ] \) denotes the greatest integer function, we will evaluate its behavior at critical points, specifically at \( x = 0 \) and \( x = 1 \).

### Step 1: Evaluate \( f(x) \) for \( x < 0 \)
For \( x < 0 \):
- \( |x| = -x \)
- \( [x-1] \) will be the greatest integer less than \( x-1 \), which is \( [x-1] = -2 \) (since \( x-1 < -1 \) for \( x < 0 \)).

Thus, for \( x < 0 \):
\[ f(x) = -x - 2 \]

### Step 2: Evaluate \( f(x) \) for \( 0 \leq x < 1 \)
For \( 0 \leq x < 1 \):
- \( |x| = x \)
- \( [x-1] = -1 \) (since \( x-1 < 0 \)).

Thus, for \( 0 \leq x < 1 \):
\[ f(x) = x - 1 \]

### Step 3: Evaluate \( f(x) \) for \( x \geq 1 \)
For \( x \geq 1 \):
- \( |x| = x \)
- \( [x-1] = 0 \) (since \( x-1 \geq 0 \)).

Thus, for \( x \geq 1 \):
\[ f(x) = x + 0 = x \]

### Step 4: Determine continuity at \( x = 0 \)
To check continuity at \( x = 0 \), we need to evaluate the left-hand limit and right-hand limit.

- **Left-hand limit** as \( x \to 0^- \):
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x - 2) = -0 - 2 = -2 \]

- **Right-hand limit** as \( x \to 0^+ \):
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x - 1) = 0 - 1 = -1 \]

Since the left-hand limit (-2) and right-hand limit (-1) are not equal, \( f(x) \) is discontinuous at \( x = 0 \).

### Step 5: Determine continuity at \( x = 1 \)
To check continuity at \( x = 1 \), we again evaluate the left-hand limit and right-hand limit.

- **Left-hand limit** as \( x \to 1^- \):
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x - 1) = 1 - 1 = 0 \]

- **Right-hand limit** as \( x \to 1^+ \):
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} x = 1 \]

Since the left-hand limit (0) and right-hand limit (1) are not equal, \( f(x) \) is discontinuous at \( x = 1 \).

### Conclusion
The function \( f(x) \) is discontinuous at both \( x = 0 \) and \( x = 1 \).

### Final Answer
The function \( f(x) \) is neither continuous at \( x = 0 \) nor at \( x = 1 \).

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