Like @Art said, I think the key thing to realise is that the elasticity is only well defined for small intervals. When you have a big change in price and quantity, the statement of "revenue increases when price decreases if demand is elastic" are no longer accurate / well defined.
We can see the problem mathematically. Let R, P, Q be the old revenue, price, and quantity respectively. Let R' be the new revenue, and $\Delta P$, $\Delta Q$ to be the change in price and quantity, respectively. Let $\epsilon_D$ be the price elasticity of demand.
\begin{align}
R & \equiv P Q \\
R' & \equiv (P + \Delta P)(Q + \Delta Q) \\
\frac{R'}{R} & = 1 + \frac{\Delta Q}{Q} + \frac{\Delta P}{P} + \frac{\Delta P \Delta Q}{PQ} \\
\end{align}
Meanwhile,
\begin{align}
\epsilon_D & \equiv \frac{\Delta Q / Q}{\Delta P / P} \leqslant 0 \\
\frac{\Delta Q}{Q} & = \frac{\Delta P}{P} \epsilon_D
\end{align}
Substituting:
\begin{align}
\frac{R'}{R} & = 1 + \frac{\Delta P}{P}(\epsilon_D+1)+(\frac{\Delta P}{P})^2\epsilon_D
\end{align}
Suppose the price decreased and the revenue also decreased. The textbooks say this means demand is inelastic. What do maths say about $\epsilon_D$?
\begin{align}
-1 < \frac{\Delta P}{P} & < 0 \\
\frac{R'}{R} & < 1 \\
\frac{\Delta P}{P}(\epsilon_D+1)+(\frac{\Delta P}{P})^2\epsilon_D & < 0 \\
\epsilon_D+1+\frac{\Delta P}{P}\epsilon_D & > 0 \\
\epsilon_D(1+\frac{\Delta P}{P}) & > -1 \\
\epsilon_D & > -\frac{1}{1+\frac{\Delta P}{P}} \\
\epsilon_D & > -\frac{P}{P+\Delta P} \neq -1
\end{align}
Suppose the price changed from 11 to 10. This means the elasticity could have any value that's greater than -1.1. If the elasticity is -1.05, which means demand is technically elastic, the revenue would still decrease in this scenario.
Notice that such discrepancy is reconciled when the changes are small:
\begin{align}
\lim_{\Delta P \rightarrow 0} (-\frac{P}{P+\Delta P}) = -1
\end{align}
What does this say about real world?
- We will never have a smooth, true-to-reality mathematical expression of the demand curve irl.
- We will never change the price by an infinitesimal amount irl.
- We likely don't care if the total revenue stays exactly the same. As long as we have a good idea of which direction it's going.
- If the demand is very elastic, the total revenue increases when you decrease the price. Vice versa.
- When the demand elasticity is roughly -1, the total revenue will stay roughly the same when you change the price.