If `f(x)=|x-1|.([x]=[-x]),` then (where [.] represents greatest integer function)

Options

A. f(a) is continuous and differentiable at x = 1
B. f(x) is discontinuous at x = 1
C. f(x) is continuous at x = 2
D. f(x) is continuous but non-differentiable at x= 1

Explanation

To solve the problem, we need to analyze the function \( f(x) = |x - 1| \cdot [x] \), where \( [x] \) denotes the greatest integer function (also known as the floor function). We will determine the continuity and differentiability of \( f(x) \) at the point \( x = 1 \).

### Step 1: Understand the components of the function

The function \( f(x) \) is composed of two parts:
1. \( |x - 1| \): This is the absolute value function, which is continuous everywhere.
2. \( [x] \): This is the greatest integer function, which is discontinuous at integer values.

### Step 2: Analyze the behavior of \( f(x) \) around \( x = 1 \)

We need to evaluate the function \( f(x) \) at points around \( x = 1 \) to check for continuity and differentiability.

- For \( x < 1 \) (for example, \( x = 0.5 \)):
\[ f(0.5) = |0.5 - 1| \cdot [0.5] = | -0.5 | \cdot 0 = 0 \]

- For \( x = 1 \):
\[ f(1) = |1 - 1| \cdot [1] = 0 \cdot 1 = 0 \]

- For \( x > 1 \) (for example, \( x = 1.5 \)):
\[ f(1.5) = |1.5 - 1| \cdot [1.5] = |0.5| \cdot 1 = 0.5 \]

### Step 3: Check the limits as \( x \) approaches 1

Now, we check the left-hand limit and right-hand limit at \( x = 1 \):

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

- Right-hand limit as \( x \to 1^+ \):
\[ \lim_{x \to 1^+} f(x) = 0.5 \]

### Step 4: Determine continuity at \( x = 1 \)

Since the left-hand limit and right-hand limit at \( x = 1 \) are not equal:
\[ \lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x) \]
Thus, \( f(x) \) is **not continuous** at \( x = 1 \).

### Step 5: Check differentiability at \( x = 1 \)

Since \( f(x) \) is not continuous at \( x = 1 \), it cannot be differentiable at that point.

### Conclusion

The function \( f(x) \) is not continuous at \( x = 1 \) and is therefore also not differentiable at \( x = 1 \).

### Final Answer

- **Continuity**: \( f(x) \) is not continuous at \( x = 1 \).
- **Differentiability**: \( f(x) \) is not differentiable at \( x = 1 \).

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