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So I have to show that this function is continuous for $a\in\mathbb{R}$, where $a>0$ and $n\in\mathbb{N}$. For some reason this seems so obvious but I don't know how to proof it. My idea was to show that $nx^n$, $-x^n$, $a$ and $nx^{n-1}$ are all continuous as that would proof that the whole function is continuous. I would really appreciate if someone could help me with this. I also need to find a fixed point of the function which would be for $x=2$, $a=16$, $n=4$ but I am not sure if that's correct.

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Write:

$$\Phi(x)=\frac{n-1}{n}x+\frac{a}{nx^{n-1}}$$

And see that both terms are continuous functions, once $x\ne 0$.

For the fixed point ($\Phi(x)=x$) we have:

$$(n-1)x^n+a=nx^n → x^n=a$$

So, if $n=4$ and $a=16$ then $x=2$ is a fixed point.

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    thanks already, but how can I proof that both terms are continuous functions? and do I have to proof that its also continuous for x=0? – 2017-01-23
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    You can't prove that it is continuous at $x=0$ because $0$ is not in the domain. – 2017-01-23
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    For prove that those terms are continuous you can use two standard approaches. 1) $ \lim_{x→x_0} f(x)=f(x_0)$. 2) use $\epsilon$ and $\delta$ approach. – 2017-01-23
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    @user406473: is it clear? – 2017-01-23