Stuff I've tried:
\begin{align}
|x^{\frac 1 n}-a^{\frac 1 n}| &< \epsilon
\\
|(x^{\frac 1 n}-a^{\frac 1 n})^n| &< \epsilon ^n
\\
\left|\sum_{i=0}^{n}C(n,i)x^{\frac in}(-a)^{1-\frac i n}\right| &< \epsilon ^n
\end{align}
(it looks like a one but the power on $x$ and $a$ it's actually $\frac i n$)
if we take $\delta \leq 1$
$$
|x| < |a| + 1
\\
\left|\sum_{i=0}^{n}C(n,i)x^{\frac in}(-a)^{1-\frac i n}\right| \leq 2^n(|a| + 1)
$$
I would love to be able to say that $\delta = \min \left\{1, \frac \epsilon{2^n(|a|+1)} \right\}$ but this doesn't seem to be the case.
I also tried using the $$x^n – y^n = (x-y)(x^{n-1} + … + y^{n-1})$$ formula but it didn't help much either.
The way I used this formula for $n=3$ for example was:
$$|\sqrt[3]x – \sqrt[3]a| < \epsilon
\\
|x-a| < \epsilon ( \sqrt[3]{x^2} + \sqrt[3]xa + \sqrt[3]{a^2})
$$
if we use the same trick $\delta \leq 1$
$$( \sqrt[3]{x^2} + \sqrt[3]xa + \sqrt[3]{a^2}) < 3\sqrt[3]{(|a|+1)^2}$$
but even if we set $\delta = \min \left\{1, 3\epsilon \sqrt[3]{(|a|+1)^2}\right\}$
it can't be shown that $|x-a|< \delta$ implies $|\sqrt[3]x – \sqrt[3]a| < \epsilon$
Best Answer
Since you've given a general $n$th root, I'm asssuming $a\ge 0$. If $a = 0$, continuity follows from the fact that for every $\epsilon > 0$, setting $\delta = \epsilon^n$ forces $|x^{1/n}| < \epsilon$ whenever $|x| < \delta$. Now suppose $a > 0$. Let $\epsilon > 0$. Let $\epsilon'$ be such that $0 < \epsilon' < \min\{a^{1/n}, \epsilon\}$. Set $\delta = \min\{(a^{1/n} + \epsilon')^n - a, a - (a^{1/n} - \epsilon')^n\}$. Then $$a - \delta > a - [a - (a^{1/n} - \epsilon')^n] = (a^{1/n} - \epsilon')^n$$ and $$a + \delta < a + [(a^{1/n} + \epsilon')^n - a] = (a^{1/n} + \epsilon')^n.$$ So for all $x$, $|x - a| < \delta$ implies $(a^{1/n} - \epsilon')^n < a - \delta < x < a + \delta < (a^{1/n} + \epsilon')^n$, which implies $a^{1/n} - \epsilon' < x^{1/n} < a^{1/n} + \epsilon'$, i.e., $|x^{1/n} - a^{1/n}| < \epsilon'\le\epsilon$. Since $\epsilon$ was arbitrary, the result follows.