Question

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 $f\left(2\right)=196$, then $f\left(1.5\right)=$

Answer 1

Since $f\left(1\right)$ is a rational number and f is a continuous function, f will takes all values between$f\left(1\right)$ and $f\left(2\right)=196$.
When $f\left(1\right)\ne 196$.
There are infinitely many irrational numbers between$f\left(1\right)$ and $196$. so
the only possibility is that f is a constant function. In this case $f\left(1.5\right)=f\left(2\right)=196.$