Question

Let $f\left(x\right)$ be defined for all $x>0$ and be continuous.Let$f\left(x\right)$ satisfy $f\left(x/y\right)=f\left(x\right)-f\left(y\right)$ for all $x,y$ and $f\left(e\right)=1$. Then which of the following may be true

• A. $f\left(x\right)$ is bounded
• B. $f\left(1/x\right)\to 0$ as $x\to \infty +$
• C. $xf\left(x\right)\to 1$ as $x\to 0+$
• D. $f^{\prime }\left(x\right)=1/x$

Answer 1

The correct option is D $f^{\prime }\left(x\right)=1/x$

A function satisfying the given functional equation is $f\left(x\right)=\log \left(x\right)$.

Clearly $f\left(x\right)$ is not bounded

and $f\left(\frac{1}{x}\right)=-\log x\nrightarrow 0$ as $x\to \infty$

Also $xf\left(x\right)=x\log x\nrightarrow 1$ as $x\to 0$

But $f^{\prime }\left(x\right)=\frac{1}{x}$

In fact, it can be shown than any continuous functions satisfying the given functional equation is of the form $k\log x$ for some $k>0$

Since $f\left(e\right)=1$ so $k=1$