Let f(x) be a continuous function defined for $1 \leq x \leq 3$. If f(x) takes rational values for all x and f(2) = 10, then the value of f(4) is

10

We know that a continuous function defined on a closed interval attains every value between its minimum and maximum values in the interval. Therefore, f(x) being continuous on [1, 3] will attain every value between its maximum (M) and minimum (m) values.

It is given that f(x) takes rational values for all x and there are infinitely many irrational values between m and M. Therefore, f(x) can take rational values for all x, if f(x) has a constant rational value at all points between x = 1 and x = 3. In other words,

f(x) = constant for all x ∈ [1, 3]

But, f(2) = 10

∴  f(x) = 10        for all x ∈ [1, 3]

⇒ f(4) = 10