Let $f\left(x\right)$ be a continuous function defined for $1\le x\le 3$. If $f\left(x\right)$ takes rational values for all $x$ and $f\left(2\right)=10$, then find the value of $f(1.5)$.

Given  $f\left(x\right)$ is continuous in $[1,3], f\left(x\right)$ will attain all values between $f\left(1\right)$ and $f\left(3\right)$. 

As, $f\left(x\right)$ takes rational values for all $x$ and there are innumerable irrational values between $f\left(1\right)$ and $f\left(3\right)$ which implies that $f\left(x\right)$ can take rational values for all $x$, if $f\left(x\right)$ has a constant rational value at all points between $x=1$ and $x=3$.

So, $f\left(2\right)=f(1.5)=10$.

We can say this by Continuity of composite function.

Continuity of composite function: If the function $u=f\left(x\right)$ is continuous at the point $x=a$ and the function $y=g\left(u\right)$ is continuous at the point $u=f\left(a\right)$, then the                                                                    composite function $y=\left(gof\right)\left(x\right)=g\left(f\right(x\left)\right)$ is continuous at the point $x=a$.

Therefore, the correct answer is option $1. f(1.5)=10$.