Let f be a continuous function on [1,3] which takes rational values for all x. If f(2) =10 then f(2.5) is equal to:

Options

A. 25
B. 20
C. `(f(1) + f(3))/2`
D. 10

Answer

To solve the problem, we need to analyze the given information about the function \( f \).

### Step-by-Step Solution:

1. **Understanding the Function**:
We are given that \( f \) is a continuous function on the interval \([1, 3]\) and takes rational values for all \( x \) in that interval.

2. **Given Value**:
We know that \( f(2) = 10 \).

3. **Properties of Rational and Irrational Numbers**:
Since \( f \) is continuous and takes rational values, we need to consider the implications of continuity. Between any two rational numbers, there exists an irrational number. If \( f \) were to take different rational values at different points, it would imply that there are irrational values in between, which would break the continuity of \( f \).

4. **Conclusion about the Function**:
The only way for \( f \) to remain continuous and take only rational values is if \( f \) is a constant function. This means that \( f(x) = k \) for some rational number \( k \) for all \( x \) in \([1, 3]\).

5. **Finding the Constant Value**:
Since we know \( f(2) = 10 \), we can conclude that \( k = 10 \). Therefore, the function must be \( f(x) = 10 \) for all \( x \) in the interval \([1, 3]\).

6. **Evaluating \( f(2.5) \)**:
Since \( f(x) = 10 \) for all \( x \), we can directly find:
\[ f(2.5) = 10 \]

### Final Answer:
Thus, \( f(2.5) = 10 \).

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