Does there always exist a consumption bundle at which the indirect utility function is the inverse of the expenditure function? Two questions:


Given $v(\vec{p},m)$ and $e(\vec{p},\bar{U})$, is there only a single point at which these are inverses of each other?  
Does an inverse always exist for a given price vector $\vec{p}$, income $m$ and $\bar{U}$, $v$ and $e$? 


These are points my mates and I are arguing about as we prepare for our midterm. 

Also, I prefer rigorous answers so if you can, feel free to be as thorough as you want.
Regarding question 1:

From the assumption that the consumer has a single income value $m$, there can only be one value for $m$. So by the notion that $e$ and $v$ are inverses, there can only be one value $e$ for which $m$ corresponds to and thus only one $u$. So there is only one point. 

Regarding Question 2: 

If we have (1) a utility function that is continuous and locally non-satiated and (2) if $m &gt; 0$ and (3) if both UMP and EMP exist, then they are equivalent. And $e$ and $v$ are inverses. 

Claim 1: Solving UMP solves EMP. 

Proof:  

Suppose bundle $c^*$ solves the UMP but not EMP. Let $c'$ solve EMP. Then (1) amount spent on $c^*$ greater then that on $c'$ and so $u(c') \geq u(c^*) $ because obviously spending more means worse off or same utility. But (2) by local nonsatiation a consumption there exists a bundle $c''$ close enough to $c'$ such that amount spent on $c''$ is less than amount spent on $c^*$ and $u(c'') &gt; u(c^*)$. (3) Contradiction because we assumed $c^*$ solves UMP. 

Claim 2: Solving EMP solves UMP. 

Proof: 

Suppose bundle $c^*$ solves EMP but not UMP. Let $c'$ solve UMP. Then (1) $u(c') &gt; u(c^*)$ even though we spend the same amount on each. That is, $c^*$ doesn't solve UMP since less utility for same $m$.   Because amount spent positive, we can find $t$ where $0&lt;t&lt;1$ such that amount spent on $tc'$ is lower than that spent on $c^*$ yet $u(tc') &gt; u(c^*)$ because $c^*$ does not solve the UMP. (3) Contradiction because we said $c^*$ solved EMP.

Summary: 


If solutions exist either to UMP or EMP, then solutions exist for the other and they are inverse functions. 
If an inverse exists, it must be a point.