Formal proof function is continuous

Question

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

I think maybe look at the proof that if $f$ and $g$ are continuous, then so is $f \cdot g$. — Trevor Gunn, Oct 17 '19 at 21:24
Thank you guys for your insight. Totally forgot about the property mentioned by @TrevorGunn . I think i'll go that way. — Gerardo Hernández, Oct 17 '19 at 21:29

score 2 · Answer 1 · answered Oct 17 '19 at 21:34

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 $3|x + y| \cdot |(x^2 + 1)^2 + (x^2 + 1)(y^2 + 1) + (y^2 + 1)^2|$.

Formal proof function is continuous

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