If `f(x) = [x^2] + sqrt({x}^2`, where [] and {.} denote the greatest integer and fractional part functions respectively,then

Options

A. f(x) is continuous at all integer points
B. f(x) is continuous and differentiable at x=0
C. f(x) is continuous for all `x in Z-(1)`
D. f(x) is not differerntiable on Z

Explanation

To solve the problem involving the function \( f(x) = [x^2] + \sqrt{\{x\}^2} \), where \([]\) denotes the greatest integer function and \(\{\}\) denotes the fractional part function, we will analyze the components of the function step by step.

### Step 1: Understand the components of the function
- The greatest integer function \([x^2]\) gives the largest integer less than or equal to \(x^2\).
- The fractional part function \(\{x\}\) is defined as \(\{x\} = x - [x]\), which represents the non-integer part of \(x\).

### Step 2: Analyze \(f(x)\)
The function can be rewritten as:
\[ f(x) = [x^2] + \sqrt{(x - [x])^2} \]
Since \(\{x\} = x - [x]\), we can express the fractional part squared:
\[ \sqrt{\{x\}^2} = |x - [x]| = \{x\} \]
Thus, we can simplify \(f(x)\) to:
\[ f(x) = [x^2] + \{x\} \]

### Step 3: Evaluate \(f(x)\) for different intervals of \(x\)
1. **For \(x\) in the interval \([n, n+1)\)** where \(n\) is an integer:
- Here, \([x] = n\) and \(\{x\} = x - n\).
- Therefore, \(f(x) = [x^2] + (x - n)\).

2. **Determine \([x^2]\)**:
- If \(n^2 \leq x^2 < (n+1)^2\), then \([x^2] = k\) where \(k\) is the largest integer satisfying \(k \leq x^2 < k+1\).

### Step 4: Check continuity and differentiability
To check if \(f(x)\) is continuous and differentiable, we need to check the points where \(x\) is an integer, as these are the points where the greatest integer function changes value.

1. **At integer points**:
- For \(x = n\), \(f(n) = [n^2] + 0 = n^2\).
- For \(x\) approaching \(n\) from the left, \(f(n^-) = [n^2] + (n - n) = n^2\).
- For \(x\) approaching \(n\) from the right, \(f(n^+) = [n^2] + (n - n) = n^2\).

Since \(f(n^-) = f(n) = f(n^+)\), \(f(x)\) is continuous at integer points.

2. **Differentiability**:
- The function \(f(x)\) is not differentiable at integer points because the derivative of the greatest integer function is not defined at those points.

### Final Result
The function \(f(x) = [x^2] + \{x\}\) is continuous everywhere but not differentiable at integer points.