If `[.]` and `{.}` denote greatest and fractional part functions respectively and `f(x)={(x((e^([x]+|x|)-2)/([x]+{2x})),x!=0),(-1,x=0):}` then (A) `f(x)` is differentiable every where (B) `f(x)` is continuous at `x=0` (C) `f(x)` is continuous every where (D) `f(x)` is not differentiable at `x=0`

Options

A. `f(x)` is differentiable every where
B. `f(x)` is continuous at `x=0`
C. `f(x)` is continuous every where
D. `f(x)` is not differentiable at `x=0`

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} \frac{x \left( e^{[x] + |x|} - 2 \right)}{[x] + \{2x\}}, & x \neq 0 \\ -1, & x = 0 \end{cases} \]

where \([x]\) denotes the greatest integer function and \(\{x\}\) denotes the fractional part function.

### Step 1: Check Continuity at \( x = 0 \)

To determine if \( f(x) \) is continuous at \( x = 0 \), we need to check if:

\[ \lim_{x \to 0} f(x) = f(0) \]

Given that \( f(0) = -1 \), we need to calculate the limit as \( x \) approaches 0.

### Step 2: Calculate the Limit as \( x \to 0 \)

For \( x \) approaching 0 (from both sides), we have:

- \([x] = 0\) since \( x \) is approaching 0.
- \(|x| = x\) when \( x \) approaches 0 from the right and \(|x| = -x\) when approaching from the left.
- \(\{2x\} = 2x\) when \( x \) is close to 0.

Thus, we can rewrite \( f(x) \) for \( x \neq 0 \):

\[ f(x) = \frac{x \left( e^{0 + |x|} - 2 \right)}{0 + \{2x\}} = \frac{x \left( e^{|x|} - 2 \right)}{2x} \]

This simplifies to:

\[ f(x) = \frac{e^{|x|} - 2}{2} \]

### Step 3: Evaluate the Limit

Now we evaluate the limit as \( x \to 0 \):

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{e^{|x|} - 2}{2} \]

As \( x \to 0 \), \( e^{|x|} \to e^0 = 1 \):

\[ \lim_{x \to 0} f(x) = \frac{1 - 2}{2} = \frac{-1}{2} \]

### Step 4: Compare the Limit with \( f(0) \)

We found that:

\[ \lim_{x \to 0} f(x) = -\frac{1}{2} \quad \text{and} \quad f(0) = -1 \]

Since the limit does not equal the function value at \( x = 0 \):

\[ \lim_{x \to 0} f(x) \neq f(0) \]

### Conclusion on Continuity

Thus, \( f(x) \) is not continuous at \( x = 0 \).

### Step 5: Differentiability at \( x = 0 \)

Since \( f(x) \) is not continuous at \( x = 0 \), it cannot be differentiable at that point.

### Final Answer

The correct option is:
(D) \( f(x) \) is not differentiable at \( x = 0 \).