Properties of Real Powers of Nonnegative Real Numbers

In the setting of The Real Numbers: Standing Notation and Background, write $\mathbb{R}_{+}=\{t\in\mathbb{R}:0\le t\}$ for the set of nonnegative real numbers, and for $t\in\mathbb{R}_{+}$ and a positive real number $a$ let $t^{a}$ be the power of $t$ with exponent $a$. The restriction of the metric $d_{\mathbb{R}}$ to $\mathbb{R}_{+}$ is a metric on $\mathbb{R}_{+}$, since the four conditions of Metric Space hold for all real numbers and hence for all nonnegative ones; it is written $d_{+}$, and $\mathbb{R}_{+}$ is regarded as the metric space $(\mathbb{R}_{+},d_{+})$. Let $a$ and $b$ be positive real numbers and let $s,t\in\mathbb{R}_{+}$. Then the following hold.

1. (Values)¶ $0\le t^{a}$; moreover $t^{a}=0$ if and only if $t=0$, so $0<t^{a}$ whenever $0<t$.

2. (Agreement with earlier powers)¶ $t^{1}=t$. For every $n\in\mathbb{N}$ the power $t^{n}$, formed with the real number $n$ as exponent, equals the natural power of $t$ with exponent $n$. Finally $t^{1/2}=\sqrt{t}$, the nonnegative square root of $t$.

3. (Multiplicativity)¶ $(st)^{a}=s^{a}t^{a}$.

4. (Exponent laws)¶ $t^{a+b}=t^{a}t^{b}$ and $(t^{a})^{b}=t^{ab}$.

5. (Monotonicity)¶ If $s\le t$ then $s^{a}\le t^{a}$, and if $s<t$ then $s^{a}<t^{a}$.

6. (Inversion and comparison)¶ The map $P_{a}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ given by $P_{a}(t)=t^{a}$ is a bijection of $\mathbb{R}_{+}$ onto $\mathbb{R}_{+}$, with inverse $P_{1/a}$; that is, $(t^{a})^{1/a}=t$ and $(t^{1/a})^{a}=t$. Consequently

7. (Continuity)¶ The map $P_{a}$ is continuous from $(\mathbb{R}_{+},d_{+})$ to $(\mathbb{R},d_{\mathbb{R}})$. Equivalently, whenever $(t_{m})_{m\in\mathbb{N}}$ is a sequence in $\mathbb{R}_{+}$ converging to $t\in\mathbb{R}_{+}$, the sequence $(t_{m}^{a})_{m\in\mathbb{N}}$ converges to $t^{a}$.