Let $f : \mathbb{R} → \mathbb{R}$ be continuous, with $f(x)f(f(x)) = 1$ for all $x ∈ \mathbb{R}$. If $f(1000) = 999$, find $f(500)$.

Question

Let $f : \mathbb{R} → \mathbb{R}$ be continuous, with $f(x)f(f(x)) = 1$ for all $x ∈ \mathbb{R}$. If $f(1000) = 999$, find $f(500)$.

I try to solve this problem, but I don't know how to use de continuity on $f$. Is anyone could give me a little hint on the continuity? And please, don't give me the answer to the question.

score 8 · Accepted Answer · answered Oct 04 '15 at 17:06

8

Hint 1: What can you say about $f(y)$ if $y$ is in the image of $f$?

Hint 2: The intermediate value theorem will be helpful. Note that $999$ is in the image of $f$.

answered Oct 04 '15 at 17:06

Ligo

660

I suspected that I could use the intermediate value theorem, but I do not know how. Thank you, I'll think now. – Oct 04 '15 at 17:08

score 3 · Answer 2 · answered Oct 04 '15 at 17:08

3

Hint Use Intermediate Value Theorem between $\frac{1}{999}$ and $999$.

$500$ belongs in this interval.

answered Oct 04 '15 at 17:08

gamma

924

Let $f : \mathbb{R} → \mathbb{R}$ be continuous, with $f(x)f(f(x)) = 1$ for all $x ∈ \mathbb{R}$. If $f(1000) = 999$, find $f(500)$.

2 Answers2

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