If `lim_(x to a) f(x)=lim_(x to a) [f(x)] and f(x) ` is non-constant continuous function, where [.] denotes the greatest integer function, then

Options

A. `underset(x to a)lim` f(x) is an integer
B. `underset(x to a)lim` f(x) is not an integer
C. f(x) has a local maximum at x=a
D. f(x) has a local minimum at x=a

Explanation

To solve the problem, we need to analyze the given limits involving the function \( f(x) \) and the greatest integer function \( [f(x)] \).

### Step-by-Step Solution:

1. **Understanding the Given Limits:**
We are given that:
\[ \lim_{x \to a} f(x) = \lim_{x \to a} [f(x)] \]
where \( f(x) \) is a non-constant continuous function and \( [.] \) denotes the greatest integer function.

2. **Rearranging the Equation:**
We can rearrange the equation as follows:
\[ \lim_{x \to a} f(x) - \lim_{x \to a} [f(x)] = 0 \]
This implies:
\[ \lim_{x \to a} (f(x) - [f(x)]) = 0 \]

3. **Taking the Limit:**
Since both limits exist, we can take the limit inside:
\[ f(a) - [f(a)] = 0 \]
This means:
\[ f(a) = [f(a)] \]
Hence, \( f(a) \) must be an integer.

4. **Analyzing the Continuity of \( f(x) \):**
Since \( f(x) \) is a non-constant continuous function, it cannot be constant over any interval. Thus, it must either be increasing or decreasing.

5. **Considering Local Maxima and Minima:**
- If \( f(x) \) has a local maximum at \( x = a \), then:
\[ \lim_{x \to a^-} f(x) \geq f(a) \quad \text{and} \quad \lim_{x \to a^+} f(x) \leq f(a) \]
However, this would imply that \( [f(x)] \) would not equal \( f(x) \) at points close to \( a \), leading to a contradiction since \( f(x) \) is continuous.
- If \( f(x) \) has a local minimum at \( x = a \), then:
\[ \lim_{x \to a^-} f(x) \leq f(a) \quad \text{and} \quad \lim_{x \to a^+} f(x) \geq f(a) \]
In this case, \( f(x) \) can equal \( [f(x)] \) for values approaching \( a \), maintaining continuity.

6. **Conclusion:**
Therefore, \( f(a) \) must be an integer, and \( f(x) \) can only have a local minimum at \( x = a \). Thus, the conclusion is that \( f(a) \) is an integer and \( f(x) \) has a local minimum at \( x = a \).

### Final Answer:
- \( f(a) \) is an integer.
- \( f(x) \) has a local minimum at \( x = a \).