Let f(x) be a continuous function defined on `[1, 3]`. If f(x) takes only rational values for all x and `f(2)=10`, then `f(2.5)=`

Options

A. 7.5
B. 12.5
C. 10
D. 15

Answer

To solve the problem, we need to analyze the properties of the function \( f(x) \) given in the question.

### Step-by-Step Solution:

1. **Understanding the Function**:
We know that \( f(x) \) is a continuous function defined on the closed interval \([1, 3]\). This means that for every \( x \) in this interval, the function does not have any jumps or breaks.

2. **Rational Values**:
The function \( f(x) \) takes only rational values for all \( x \) in the interval. We are given that \( f(2) = 10 \), which is a rational number.

3. **Continuity Implication**:
Since \( f(x) \) is continuous on \([1, 3]\) and takes only rational values, we must consider the implications of continuity. A continuous function that only takes rational values must be constant. This is because between any two rational numbers, there are infinitely many irrational numbers, and a continuous function that jumps from one rational value to another would have to take on irrational values in between, which contradicts the condition that \( f(x) \) is only rational.

4. **Constant Value**:
Given that \( f(2) = 10 \) and \( f(x) \) is constant, it follows that \( f(x) = 10 \) for all \( x \) in the interval \([1, 3]\).

5. **Finding \( f(2.5) \)**:
Since \( f(x) = 10 \) for all \( x \) in \([1, 3]\), we can directly substitute \( x = 2.5 \):
\[ f(2.5) = 10 \]

### Conclusion:
Thus, the value of \( f(2.5) \) is \( 10 \).

### Final Answer:
\[ f(2.5) = 10 \]

---