Let f be a function defined on R by `f(x)=[x] + sqrt(x-[x])` then

Options

A. f is not continuous at every `x in I`
B. f is not continuous at every `x in R ~ I`
C. f is a continuous function
D. none of these

Answer

To determine the continuity of the function \( f(x) = [x] + \sqrt{x - [x]} \), where \([x]\) is the greatest integer function, we will analyze the function at integer points and non-integer points.

### Step 1: Understanding the Function
The function \( f(x) \) can be expressed as:
- \([x]\) is the greatest integer less than or equal to \( x \).
- \( \sqrt{x - [x]} \) is the square root of the fractional part of \( x \).

### Step 2: Check Continuity at Integer Points
Let \( n \) be an integer. We will check the left-hand limit (LHL) and the right-hand limit (RHL) at \( n \).

#### Left-Hand Limit (LHL)
\[ \lim_{x \to n^-} f(x) = \lim_{x \to n^-} \left( [x] + \sqrt{x - [x]} \right) \]
For \( x \) approaching \( n \) from the left, \([x] = n - 1\) and \( x - [x] \) approaches \( 1 \):
\[ \lim_{x \to n^-} f(x) = (n - 1) + \sqrt{1} = n - 1 + 1 = n \]

#### Right-Hand Limit (RHL)
\[ \lim_{x \to n^+} f(x) = \lim_{x \to n^+} \left( [x] + \sqrt{x - [x]} \right) \]
For \( x \) approaching \( n \) from the right, \([x] = n\) and \( x - [x] \) approaches \( 0 \):
\[ \lim_{x \to n^+} f(x) = n + \sqrt{0} = n + 0 = n \]

### Step 3: Value of the Function at Integer Points
\[ f(n) = [n] + \sqrt{n - [n]} = n + 0 = n \]

### Step 4: Conclusion at Integer Points
Since:
- \( \lim_{x \to n^-} f(x) = n \)
- \( \lim_{x \to n^+} f(x) = n \)
- \( f(n) = n \)

Thus, \( f(x) \) is continuous at every integer \( n \).

### Step 5: Check Continuity at Non-Integer Points
For \( x \) that is not an integer, \([x]\) remains constant in the interval \((n, n+1)\) for any integer \( n \). Therefore, \( f(x) \) is continuous in these intervals as well since both components of \( f(x) \) are continuous.

### Final Conclusion
Since \( f(x) \) is continuous at all integer points and continuous in the intervals between integers, we conclude that \( f(x) \) is continuous everywhere on \( \mathbb{R} \).

Thus, the correct option is:
- **f is continuous everywhere.**