How do I prove, using the definition, that the nth root is a continuous function? 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 $|\sqrt[3]x - \sqrt[3]a| < \epsilon$
Prove continuity for cubic root using epsilon-delta

With some accessible work, you can also show the inequality
$$
\left|x^{1/n}-y^{1/n}\right|\le|x-y|^{1/n}
$$
showing Hölder continuity and uniform continuity on $[0,\infty)$.

The proof starts with considering $0\le y\le x$, so that $x^{1/n}=y^{1/n}+h$, $h\ge0$ and uses only the binomial theorem,
$$
x-y=(y^{1/n}+h)^n-y=((y^{1/n})^n+...+h^n)-y\ge h^n
\\
\implies\sqrt[n]{x-y}\ge x^{1/n}-y^{1/n}
$$
