Is my Proof for Proving g is Injective on (a,b) Correct?

[K29]

Homework Statement

g is a continuous and differentiable function on(a,b). Prove that if for all x\in(a,b), g'(x)\neq0\Rightarrow g is an injective function.

Is there anything wrong about my proof? I'm not comfortable in making logical mathematical arguments: I usually make small inaccuracies. Or do a lot of work going down a bad path Any help appreciated.

The Attempt at a Solution

We select an arbitrary sub-interval on (a,b). We call it (m,n) We know that g is also continuous and differentiable on any sub-interval of the interval that it is continuous and differentiable on.

The "injective functoin" has the definition that f(a)=f(b)\Rightarrow a=b That is, no two values on the domain can map to the same point on the co-domain.

So for now we shall assume thatf(m)=f(n). This allows us to make use of the Mean value theorem which states that if a function is continuous and, differentiable on an interval, in this case (m,n) and f(m)=f(n) then \exists c\in (m,n) such that g'(c) = 0. This leads to a contradiction as g'(x)\neq0. Therefor no such interval (m,n) can exist. Hence f(m)=f(n)\Rightarrow m=n (The point m on the domain is equivalent to(the same point as) n) and g is injective.

 

[Dick]

The proof is ok. Your phrasing at the beginning is pretty awkward. You aren't selecting (m,n) to be an arbitrary sub-interval. You are doing a proof by contradiction. So you assume f is NOT injective. If f is NOT injective then there are two different points m and n such that f(m)=f(n). Now the rest of the proof is fine.

 

[K29]

Thanks very much!