The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $\exp$ be the exponential function and $\log:(0,\infty)\to\mathbb{R}$ the natural logarithm, and let $[0,\infty)$ denote the set of nonnegative real numbers, a subset of $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line. Let $\phi:[0,\infty)\to\mathbb{R}$ be the function with

1. (Exponential)¶ $\exp(u)\ge1+u$ for every $u\in\mathbb{R}$.

2. (Logarithm)¶ $1-t^{-1}\le\log t\le t-1$ for every positive $t\in\mathbb{R}$.

3. (Continuity and measurability)¶ $\phi$ is continuous on $[0,\infty)$. Consequently, for every measure space $(X,\mathcal{F},\mu)$ and every measurable $f:X\to\mathbb{R}$ with $0\le f(x)$ for every $x\in X$, the function $\phi\circ f$ is measurable.

4. (Young's inequality)¶ $a\,s\le\phi(s)+\exp(a-1)$ for every $a\in\mathbb{R}$ and every $s\in[0,\infty)$.

5. (Lower bounds)¶ $s-1\le\phi(s)$ and $-\exp(-1)\le\phi(s)$ for every $s\in[0,\infty)$.