nLab real nth root function

Idea

Given an integer $n$, The different types of $n$-th root partial functions on the real numbers that satisfy the functional equation $f(x)^n = x$ on some subset of the real numbers.

Definition

Let us define the real numbers to be a sequentially Cauchy complete Archimedean ordered field, since that is the minimum requirement for which the inverse function theorem is true.

There exists an $n$-th root function for every integer $n \in \mathbb{Z}$ apart from zero, defined on the positive real numbers by the equation:

$$0 \lt \vert n \vert \vdash \sqrt[n]{x} \coloneqq \exp(\frac{1}{n} \ln(x))$$

Now, assume that $n \gt 1$. Then one could extend the $n$-th root function to a function defined on the half-open interval $[0, \infty)$. Let us define continuous functions $f_k:[0, \infty) \to [0, \infty)$:

$$f_k(x) = \begin{cases} 1/n^k & \mathrm{when}\; x \leq 1/n^{2 k} \\ \exp(\frac{1}{n} \ln(x)) & \mathrm{when}\; x \geq 1/n^{2 k} \\ \end{cases}$$

As stated, that requires knowing whether $x \leq 1/n^{2 k}$ or $x \geq 1/n^{2 k}$, but it is possible to work around this by patching three functions together:

• $f^{0}_k:[0,1/n^{2 k}) \to \mathbb{R}$ is the linear function $f(x) = n^k x$,
• $f^{+}_k:(0,\infty) \to \mathbb{R}$ is defined as $\min(n^k x,\exp(\frac{1}{n} \ln(x)))$.

Since these functions agree on their overlap, and their domains comprise all of $[0, \infty)$ we do get a total function $f_k:[0, \infty) \to [0, \infty)$ as a result.

Now the sequence of functions $(f_k)_{k=0}^\infty$ so defined converges uniformly on any bounded interval to a continuous function $\sqrt[n]{(-)}:[0, \infty) \to [0, \infty)$ called the principal nth root function. It is easily seen that $(\sqrt[n]{x})^n = x$ and $\sqrt[n]{x^n} = x$.

Additionally, if $n$ is an odd integer greater than 1, then there exist $n$-th root functions defined on the entirety of the real numbers.

Let us patch this sequence of triples of functions together:

• $f^{-}_k:(-\infty,0) \to \mathbb{R}$ is defined as $\max(-n^k x,-\exp(\frac{1}{n} \ln(-x)))$
• $f^{0}_k:(-1/n^{2 k},1/n^{2 k}) \to \mathbb{R}$ is the linear function $f(x) = n^k x$,
• $f^{+}_k:(0,\infty) \to \mathbb{R}$ is defined as $\min(n^k x,\exp(\frac{1}{n} \ln(x)))$.

Since these functions agree on their overlap, and their domains comprise all of $\mathbb{R}$ we do get a total function $f_k:\mathbb{R} \to \mathbb{R}$ as a result.

Now the sequence of functions $(f_k)_{k=0}^\infty$ so defined converges uniformly on any bounded interval to a continuous function $\sqrt[n]{(-)}:\mathbb{R} \to \mathbb{R}$ called the nth root function for odd integer $n$. It is easily seen that $(\sqrt[n]{x})^n = x$ and $\sqrt[n]{x^n} = x$.

Other nth root functions

According to (Richman 2010), given the existence of an nth root function, there are an uncountable number of functions that satisfy the functional equation $f(x)^n = x$ on some subset of the real numbers. Each of these could be called a real “nth root function”.

For example, let $1_{\mathbb{Q}}:\mathbb{R} \to \mathbb{R}$ be the constructive Dirichlet indicator function, defined as $1_{\mathbb{Q}}(x) \coloneqq 1$ for every rational number $q \in \mathbb{Q}$, and $1_{\mathbb{Q}}(x) \coloneqq 0$ for every real number $x$ apart from every rational number $q \in \mathbb{Q}$

$$\forall q \in \mathbb{Q}. \vert x - q \vert \gt 0$$

Then the function $f(x) \coloneqq (-1)^{1_{\mathbb{Q}}(x)} \sqrt[n]{x}$ is a real nth root function, even though it is nowhere continuous, and not defined on the entire open interval $(n, \infty)$.

See also

• root
• real square root function

References

• Fred Richman (2010). Algebraic functions, calculus style. Fred Richman’s documents.