A demand curve with constant slope over all quantity values can have a continuously changing price elasticity of demand.

The price elasticity of demand measures the responsiveness of quantity demanded to a change in price, and it depends on the percentage change in quantity demanded relative to the percentage change in price. Even if a demand curve has a constant slope (indicating a linear relationship between price and quantity), the relative changes (percentages) in price and quantity can vary at different points along the curve, leading to a continuously changing price elasticity of demand.