A theory of continuous rates and applications to the theory of growth and obsolescence rates

Abstract

For functions f of a continuous variable t, we define the term ''rate'' (as, e.g., rate of growth or of obsolescence) as the exponential function of the derivative of the logarithm of this function (i.e., e(log f)'). This replaces discrete calculations, such as f(t + 1)/f(t), which is not so appropriate in this continuous context. We investigate this transformation (which is in fact the exponential function of the Fechner law), and show that it indeed has all properties that we can expect from a ''rate'' function. We then apply these findings to the results of three previous papers and again prove the main results in this continuous setting.