The correct answer is A.
The elasticity of demand is a measure of how responsive consumers are to changes in price. A straight line, downward-sloping demand curve implies that, as price falls, the elasticity of demand increases. This is because, at lower prices, consumers are more willing to buy a good, and a small change in price will lead to a larger change in quantity demanded.
Option B is incorrect because, as price falls, the elasticity of demand increases, not decreases.
Option C is incorrect because, as price falls, the elasticity of demand changes, not remains the same.
Option D is incorrect because, as price falls, the elasticity of demand is not zero.