Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm trying to solve this rather simple question, but no luck.

Let f be continuous and injective in $\mathbb R$.
prove that if $ \lim_{x\to \infty}{f(x)}=\infty $ than ${f}$ is monotonic increasing in $\mathbb R$.

My approach is to negatively assume that given some a and b such that ${a < b}$ then ${f(a) \ge f(b)}$, but this is where I got stuck. I don't know how to continue from here.

Any help would be appreciated!

share|cite|improve this question
A good start. By injectivity, $f(a) > f(b)$. Now, what does the continuity tell you about $f([b,\infty))$? – Daniel Fischer Mar 22 '14 at 11:16

As you has said, let's suppose that there exist two real numbers $a<b$ such that $f(a)\geq f(b)$. Since $f$ is injective, $f(a)>f(b)$. Now we have $\lim_{x\rightarrow\infty}=\infty$, so there exists $c>b$ such that $f(c)>f(a)$. The continuity of $f$ grants a point $d\in (b,c)$ such that $f(d)=f(a)$, a contradiction since $f$ is injective.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.