Growth Bound for a Lipschitz Function with Small Derivative

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background with the dimension $1$, throughout identifying a point of $\mathbb{R}^{1}$ with its single coordinate, so that $\mathbb{R}^{1}$ and $\mathbb{R}$ are written interchangeably; under this convention the Euclidean norm of a point is its absolute value and $d_{E}(s,t)=|s-t|$, by claims 1 and 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, so that the closed ball $\bar{B}(y,r)$ is the closed interval with endpoints $y-r$ and $y+r$. By Lebesgue Measure on $\mathbb{R}^n$ the measure $\lambda_{1}$ is the Lebesgue measure $\lambda$ on the Borel $\sigma$-algebra of $\mathbb{R}$. Accordingly $\lambda^{\ast}$ denotes Lebesgue outer measure on subsets of $\mathbb{R}$, and null, Lipschitz and the image notation $\phi(A)$ have the meanings fixed in that setting.

Let $I\subseteq\mathbb{R}$ be a nonempty open interval, let $L\in\mathbb{R}$ with $0<L$, let $\phi:I\to\mathbb{R}$ be Lipschitz with constant $L$, and let $K\in\mathbb{R}$ with $0\le K$. Derivatives are those of Derivative of a Real Function on an Interval. Then the following hold.

1. (Growth bound) (i) Let $A\subseteq I$ satisfy $\lambda^{\ast}(A)<\infty$, and suppose that at every $x\in A$ the function $\phi$ has a derivative $\phi'(x)$ with $|\phi'(x)|\le K$. Then

$$\lambda^{\ast}\bigl(\phi(A)\bigr)\le K\,\lambda^{\ast}(A).$$

2. (Increment bound from an almost everywhere derivative bound) (ii) Suppose there is a null set $N\subseteq I$ such that at every $x\in I\setminus N$ the function $\phi$ has a derivative $\phi'(x)$ with $|\phi'(x)|\le K$. Then

$$|\phi(y)-\phi(x)|\le K\,|y-x|\qquad\text{for all }x,y\in I .$$