1. ## Question about continuous function

For some continuous function $\displaystyle f:\mathbb{R}\rightarrow \mathbb{R}$, if I know the value of $\displaystyle \frac{f'(x)}{f(x)}$ at some point $\displaystyle x=a$, does that tell me any information about the value of either $\displaystyle f'(a)$ or $\displaystyle f(a)$?

I'm doubting it, but I want to make sure. Say for example I knew that this ratio was irrational. Does that tell me anything about the rationality of $\displaystyle f'(a)$ or $\displaystyle f(a)$?

2. Originally Posted by Media_Man
For some continuous function $\displaystyle f:\mathbb{R}\rightarrow \mathbb{R}$, if I know the value of $\displaystyle \frac{f'(x)}{f(x)}$ at some point $\displaystyle x=a$, does that tell me any information about the value of either $\displaystyle f'(a)$ or $\displaystyle f(a)$?
If the ratio exists, you know $\displaystyle f(a) \not= 0$.

If the ratio were irrational, then at least one of $\displaystyle f(a)\text{ or }f'(a)$ must be irrational.