# Injective

• Nov 23rd 2006, 10:49 AM
OntarioStud
Injective
Happy Thanksgiving to those of you in the States.

I have a problem:

Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
a<c<b.
Prove that f cannot be injective.

How would you work through this proof?
• Nov 23rd 2006, 11:00 AM
CaptainBlack
Can you tell us why you need three accounts?

RonL
• Nov 23rd 2006, 11:09 AM
ThePerfectHacker
Quote:

Originally Posted by OntarioStud
Happy Thanksgiving to those of you in the States.

I have a problem:

Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
a<c<b.
Prove that f cannot be injective.

How would you work through this proof?

I am going to make this proof as if it was a maximum point the minimum point is anagolous.

Assume,
$\displaystyle f:[a,b]\to \mathbb{R}$ is an injective map.
Also that the function has a maximum $\displaystyle c\in (a,b)$.
$\displaystyle [a,c],[c,b]$
Without lose of generality assume $\displaystyle f(a)\geq f(b)$. Pick any number $\displaystyle d$ such that $\displaystyle f(b) \leq d< f(c)$
$\displaystyle f(a)\leq d< f(c)$
By the intermediate value theorem there is a point $\displaystyle m_{1,2}$ such that $\displaystyle f(m_1)=d$ on $\displaystyle [a,c]$ and $\displaystyle f(m_2)=d$ on $\displaystyle [c,b]$. Thus, $\displaystyle f(m_1)=f(m_2)$, and if $\displaystyle m_1\not = m_2$ the function fails to be injective. The only way it can is when $\displaystyle m_1=m_2=c$ but that is not possible.