Usually, elasticity of a demand curve throughout its length isn't the same (Fig. below). It varies between 0 and ∞, or in other words,

0 ≤ e_p ≥ ∞

In some cases, though, the elasticity remains the same throughout the length of demand curve. Such demand curves can be placed in the subsequent categories: (i) perfectly inelastic (e = 0); (ii) unitary elastic (e = 1); and (iii) perfectly elastic (e = ∞). These three types of demand curves are demonstrated in Fig. below (a), (b) and (c), respectively.

Fig: Constant Elasticity Demand Curve