<s>[INST] Describe the condition for a curve to be rectifiable between two points. [/INST] A curve can be considered rectifiable between two points if it has a finite length. In other words, the curve can be thought of as a continuous mapping from an interval of real numbers to the plane, and its length can be defined as the supremum of the lengths of all piecewise linear approximations of the curve.

For a curve to be rectifiable, it must satisfy some additional conditions, such as being continuous and having a finite rate of change at each point along the curve. Additionally, the curve should not have any corners or cusps, as these would cause the length of the curve to become infinite.

In mathematical terms, if a curve is given by a continuous function f(t) for t in some interval [a, b], it is rectifiable if the integral of the magnitude of the derivative of f(t) with respect to t, from a to b, is finite:

∫_a^b ||f'(t)|| dt < ∞

where ||f'(t)|| is the magnitude of the derivative of f(t) at t and the integral is taken over the interval [a, b]. </s>