Question

lf $f\left(x\right)$ takes only rational values for all real $x$ and is continuous, then $f^{\prime }\left(5\right)=$

• A. $1$
• B. $5$
• C. $-1$
• D. $0$

Answer 1

The correct option is D $0$

From the number theory we know, that between any 2 rational numbers there is an irrational number.
Hence the function f(x) can be continuous, and take only rational values if and only if it is a constant function.
Thus $f\left(x\right)=c$ where c is a constant,
Hence, $f^{\prime }\left(x\right)=0$ for all x.