Prove that if f is increasing on an interval I, then f is one to one on I.

Question

How do I even begin to do this problem? I don't know where to even begin.

The professor of the class tried to give us hints (as this is a redo to our homework) and said "The contrapositive is 'If f is not one to one, then f is not increasing.' Write out carefully and clearly what it means to say that f is not one to one and that f is not increasing. Then using the hypothesis, prove f is not increasing."

I am even more lost now that he has given us this hint, any idea on how to start this proof?

Edit - I'm not asking for you guys to do the proof, I just don't know where to start and my professor doesn't answer any of my emails, so I'm turning to you guys.

score 1 · Answer 1 · answered Nov 20 '13 at 21:47

1

By definition, a (strictly) increasing function is such that $x<y\implies f(x)<f(y)$. In particular, if $x\neq y$; either $x<y$ or $y<x$. In any case, what do you conclude about $f(x)$ and $f(y)$?

answered Nov 20 '13 at 21:47

Pedro

122,002

$f(x)$ and $f(y)$ are not equal? – DChung91 Nov 20 '13 at 21:53
@DChung91 True, but why? – Pedro Nov 20 '13 at 21:54
This is where I start to fall apart...I cannot think of why. – DChung91 Nov 20 '13 at 21:59
@DChung91 For any two pair of real numbers, either $a<b,a=b$ or $b<a$. In particular, saying that $a\neq b$ is the same sa saying that either $a<b$ or $b<a$ hold true. If you assume $x<y$ and use that $f$ is increasing, what can you say about $f(x)$ and $f(y)$? – Pedro Nov 20 '13 at 22:04
If we assume that $x < y$ and use the fact that $f$ is increasing, then $f(x) < f(y)$? – DChung91 Nov 20 '13 at 22:07
@DChung91 Yes. ${}{}{}{}$ – Pedro Nov 20 '13 at 22:08

score 0 · Answer 2 · answered Nov 20 '13 at 21:44

0

Assume that $f$ is not one-to-one. Then there exist $x, y \in I$ such that $x \neq y$ and $f(x) = f(y)$..

answered Nov 20 '13 at 21:44

Tom

9,978

Using this start to the proof, do we have to mention anywhere in this that $x<y$? – DChung91 Nov 20 '13 at 22:01

Prove that if f is increasing on an interval I, then f is one to one on I.

2 Answers2