<s>[INST] Formal proof function is continuous I'm a bit stuck in this formal proof that the functions is continuos in all of it's domain.
$f(x)= 3(x^2+1)^3$
$ \epsilon >0 , \delta > 0  $
$|x-c|<\delta  \to  |3(x^2+1)^3-3(c^2+1)^3|     \to   3|x^6-c^6+3x^4+3x^2-3c^4-3c^2| $ 
that's where I get to by myself. Could somebody  help me out :) . Thanks
 [/INST] As I said in the comments: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ so we can write this as
\begin{align}
&3|(x^2 + 1) - (y^2 + 1)| \cdot |(x^2 + 1)^2 + (x^2 + 1)(y^2 + 1) + (y^2 + 1)^2| \\
={} & 3|x - y| \cdot |x + y| \cdot |(x^2 + 1)^2 + (x^2 + 1)(y^2 + 1) + (y^2 + 1)^2|.
\end{align}
Then you just need a crude bound on $3|x + y| \cdot |(x^2 + 1)^2 + (x^2 + 1)(y^2 + 1) + (y^2 + 1)^2|$.
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