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Thread: Question about continuous function

  1. #1
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    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 $\displaystyle f(a)$?
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    Quote Originally Posted by Media_Man View Post
    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.
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