Let `f(x)` be a continuous function defined for `1 <= x <= 3.` If `f(x)` takes rational values for all `x and f(2)=10` then the value of `f(1.5)` is :

Answer

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

### Step-by-Step Solution:

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

2. **Values of the Function**: The problem states that \( f(x) \) takes rational values for all \( x \) in the interval. Additionally, we are given that \( f(2) = 10 \).

3. **Finding \( f(1.5) \)**: We need to find the value of \( f(1.5) \). Since \( f(x) \) is continuous and takes rational values, we can consider the implications of these conditions.

4. **Using the Intermediate Value Theorem**: The Intermediate Value Theorem states that if \( f \) is continuous on an interval and takes two values at the endpoints of that interval, it must take every value between those two values at least once.

5. **Rational Values**: Since \( f(x) \) takes rational values and \( f(2) = 10 \), we need to consider what values \( f(x) \) can take between \( x = 1 \) and \( x = 2 \).

6. **Considering the Value at \( x = 1.5 \)**: The only way for \( f(x) \) to remain continuous and only take rational values while also being equal to 10 at \( x = 2 \) is if \( f(x) \) is constant. If \( f(x) \) were to take any other rational value at \( x = 1.5 \), it would have to jump to 10 at \( x = 2 \), which would violate the continuity of the function.

7. **Conclusion**: Therefore, since the only way to satisfy all these conditions is for \( f(x) \) to be constant, we conclude that \( f(1.5) = f(2) = 10 \).

Thus, the value of \( f(1.5) \) is \( \boxed{10} \).