If [.] denotes the greatest integer function, then `f(x)=[x]+[x+(1)/(2)]`

Options

A. is continuous at `x=(1)/(2)`
B. is discontinuous at `x=(1)/(2)`
C. `underset(x to ((1)/(2)))lim f(x)=2`
D. `underset(x to ((1)/(2))^(-))lim f(x)=1`

Explanation

To determine the continuity of the function \( f(x) = [x] + [x + \frac{1}{2}] \) at \( x = \frac{1}{2} \), we will analyze the function in the intervals around \( x = \frac{1}{2} \).

### Step 1: Define the intervals
We will evaluate the function in two intervals:
1. \( 0 \leq x < \frac{1}{2} \)
2. \( \frac{1}{2} \leq x < 1 \)

### Step 2: Evaluate \( f(x) \) for \( 0 \leq x < \frac{1}{2} \)
In this interval, the greatest integer function \( [x] \) will be:
- \( [x] = 0 \) (since \( x \) is less than \( \frac{1}{2} \))

Now, evaluate \( [x + \frac{1}{2}] \):
- Since \( x + \frac{1}{2} < 1 \) for \( x < \frac{1}{2} \), we have \( [x + \frac{1}{2}] = 0 \).

Thus, for \( 0 \leq x < \frac{1}{2} \):
\[ f(x) = [x] + [x + \frac{1}{2}] = 0 + 0 = 0 \]

### Step 3: Evaluate \( f(x) \) for \( \frac{1}{2} \leq x < 1 \)
In this interval, the greatest integer function \( [x] \) will be:
- \( [x] = 0 \) (since \( x \) is still less than \( 1 \))

Now, evaluate \( [x + \frac{1}{2}] \):
- For \( \frac{1}{2} \leq x < 1 \), \( x + \frac{1}{2} \) will be in the range \( [1, 1.5) \), thus \( [x + \frac{1}{2}] = 1 \).

Therefore, for \( \frac{1}{2} \leq x < 1 \):
\[ f(x) = [x] + [x + \frac{1}{2}] = 0 + 1 = 1 \]

### Step 4: Check continuity at \( x = \frac{1}{2} \)
To check the continuity at \( x = \frac{1}{2} \), we need to find:
- \( \lim_{x \to \frac{1}{2}^-} f(x) \)
- \( \lim_{x \to \frac{1}{2}^+} f(x) \)
- \( f\left(\frac{1}{2}\right) \)

From our evaluations:
- \( \lim_{x \to \frac{1}{2}^-} f(x) = 0 \)
- \( \lim_{x \to \frac{1}{2}^+} f(x) = 1 \)

Since these two limits are not equal:
\[ \lim_{x \to \frac{1}{2}^-} f(x) \neq \lim_{x \to \frac{1}{2}^+} f(x) \]

Thus, \( f(x) \) is not continuous at \( x = \frac{1}{2} \).

### Final Result
The function \( f(x) \) is not continuous at \( x = \frac{1}{2} \).

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