A Bounded-Derivative Truncation of the Cube Map on the Real Line

Let $\mathbb{R}$ be the real numbers, let $|\cdot|$ be the absolute value on $\mathbb{R}$, let $s^{m}$ denote the $m$-th power of $s\in\mathbb{R}$ for a natural number $m$, and write $t^{-1}$ for the multiplicative inverse of a nonzero $t\in\mathbb{R}$. Differentiability of a function on an interval at an interior point, and the derivative there, are those of that definition; by claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line the set $\mathbb{R}$ is an interval and every point of $\mathbb{R}$ is an interior point of it, so these notions apply to a function on $\mathbb{R}$ at every point, and for a function $f:\mathbb{R}\to\mathbb{R}$ differentiable at every point we write $f'$ for the function on $\mathbb{R}$ whose value at $s$ is the derivative of $f$ at $s$. We regard $\mathbb{R}$ also as the Euclidean space $\mathbb{R}^{1}$, which is open in itself by claim 1 of Euclidean Space Is Open in Itself; Euclidean continuity of a function from $\mathbb{R}$ to $\mathbb{R}$ at a point is as in Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable, being of class $C^{1}$ on $\mathbb{R}^{1}$ is as defined there, and $\partial_{1}$ denotes the partial derivative with respect to the first variable.

Let $\varepsilon\in\mathbb{R}$ satisfy $0<\varepsilon$. Then the following hold.

1. (The truncation is defined)(i) For every $s\in\mathbb{R}$ one has $1\le1+\varepsilon s^{2}$; in particular $1+\varepsilon s^{2}$ is nonzero and $0<(1+\varepsilon s^{2})^{-1}\le1$. Accordingly $\phi_{\varepsilon}$ denotes the function from $\mathbb{R}$ to $\mathbb{R}$ whose value at $s$ is

$$\phi_{\varepsilon}(s)=s^{3}\,(1+\varepsilon s^{2})^{-1}.$$

2. (Differentiability and the derivative)(ii) The function $\phi_{\varepsilon}$ is differentiable at every point of $\mathbb{R}$, its derivative function being given by

$$\phi_{\varepsilon}'(s)=(3s^{2}+\varepsilon s^{4})\,\bigl((1+\varepsilon s^{2})^{-1}\bigr)^{2}\qquad\text{for every }s\in\mathbb{R},$$

where $3$ denotes $1+1+1$. Moreover $\phi_{\varepsilon}$ is of class $C^{1}$ on $\mathbb{R}^{1}$, and

$$\partial_{1}\phi_{\varepsilon}(s)=\phi_{\varepsilon}'(s)\qquad\text{for every }s\in\mathbb{R};$$

in particular $\phi_{\varepsilon}$ and $\phi_{\varepsilon}'$ are Euclidean continuous at every point of $\mathbb{R}$.

3. (The derivative is nonnegative and bounded)(iii) For every $s\in\mathbb{R}$,

$$0\le\phi_{\varepsilon}'(s)\le4\,\varepsilon^{-1},$$

where $4$ denotes $1+1+1+1$.

4. (Sign and growth)(iv) For every $s\in\mathbb{R}$,

$$0\le s\,\phi_{\varepsilon}(s),\qquad |\phi_{\varepsilon}(s)|\le|s|^{3}.$$

5. (Approximation of the cube map)(v) For every $s\in\mathbb{R}$,

$$\bigl|\phi_{\varepsilon}(s)-s^{3}\bigr|\le\varepsilon\,|s|^{5}.$$