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Let $B=\Vert b{\Vert }_{\infty }$ and define the scale function of a one-dimensional diffusionSince $b$ is continuous, $g\in C^2$, andThus $g$ is strictly increasing. The Itô formula for the additive-noise equation givesThe integrand is locally square-integrable: a continuous path $X$ has compact range on every finite interval, where $g^{\prime }$ is bounded. HenceThis is the scale transform for an additive-noise diffusion. Its range is the open interval $I=g(\mathbb{R})$, which need not be all of $\mathbb{R}$.

The diffusion coefficient obtained from a scale transform for an additive-noise diffusion need only be defined on an open interval. If it is Lipschitz, its limit at a finite endpoint is zero: a positive limit would bound the derivative $1/h$ of the inverse scale and prevent the inverse from diverging. Extending it by zero beyond finite endpoints therefore preserves global Lipschitz continuity.