If `f(x)=|2-x|+(2+x)`, where (x)=the least integer greater than or equal to x, them

Options

A. `underset(x to 2^(-))lim f(x)=f(2)=2`
B. f(x) is continuous and differentiable at x=2
C. f(x) is neither continuous nor differentiable at x=2
D. f(x) is continuous and non-differentiable at x=2

Explanation

To solve the problem, we need to analyze the function \( f(x) = |2 - x| + \lfloor 2 + x \rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

### Step 1: Evaluate the function at \( x = 2 \)
First, we calculate \( f(2) \):
\[ f(2) = |2 - 2| + \lfloor 2 + 2 \rfloor = |0| + \lfloor 4 \rfloor = 0 + 4 = 4 \]

### Step 2: Calculate the left-hand limit as \( x \) approaches 2
Next, we find the left-hand limit \( \lim_{x \to 2^-} f(x) \):
For \( x < 2 \):
\[ f(x) = |2 - x| + \lfloor 2 + x \rfloor = (2 - x) + \lfloor 2 + x \rfloor \]
As \( x \) approaches 2 from the left, \( 2 + x \) approaches 4, which is an integer. Thus, \( \lfloor 2 + x \rfloor = 4 \) for \( x \) just less than 2.

Now substituting this into the equation:
\[ \lim_{x \to 2^-} f(x) = (2 - 2) + 4 = 0 + 4 = 4 \]

### Step 3: Compare the left-hand limit with the value of the function at \( x = 2 \)
We have:
\[ \lim_{x \to 2^-} f(x) = 4 \quad \text{and} \quad f(2) = 4 \]
Since both the left-hand limit and the function value at \( x = 2 \) are equal, we can conclude that the function is continuous at \( x = 2 \).

### Step 4: Check differentiability at \( x = 2 \)
To check differentiability, we need to consider the derivative from the left and right:
1. For \( x < 2 \):
\[ f(x) = (2 - x) + 4 \implies f'(x) = -1 \]
2. For \( x > 2 \):
\[ f(x) = |2 - x| + \lfloor 2 + x \rfloor = (x - 2) + \lfloor 2 + x \rfloor \]
As \( x \) approaches 2 from the right, \( \lfloor 2 + x \rfloor = 4 \) as well, so:
\[ f(x) = (x - 2) + 4 \implies f'(x) = 1 \]

Since the left-hand derivative \( f'(2^-) = -1 \) and the right-hand derivative \( f'(2^+) = 1 \) are not equal, \( f(x) \) is not differentiable at \( x = 2 \).

### Conclusion
- The function \( f(x) \) is continuous at \( x = 2 \) but not differentiable at that point.
- Therefore, the correct option is that \( f(x) \) is continuous and non-differentiable at \( x = 2 \).

### Final Answer
The function \( f(x) \) is continuous and non-differentiable at \( x = 2 \).

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