Unitary Elastic Demand
The way to examine such issues is by general mathematical treatment, not arithmetic examples.  

We have total revenue

$$TR = Q(P)\cdot P$$

When will Total Revenue remain constant as price changes?  

We want

$$\frac{\partial TR}{\partial P} =0 \implies \frac{\partial Q(P)}{\partial P}\cdot P + Q(P) =0$$

$$\implies  \frac{\partial Q(P)}{\partial P} = -\frac {Q(P)}{P}$$

which is to say that the price elasticity of demand is equal to minus unity.  

Is it possible for this to hold over the whole range of $P$ and not just momentarily, at a specific value of $P$?

This entails to find a solution to a simple differential equation (using generic notation)

$$y' + \frac{y}{x} = 0$$

which gives

$$Q(P) = \frac {A}{P},\;\;\; A&gt;0$$

So if the demand function is as above, it has unitary price elasticity everywhere.