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let $f: \Bbb R \to \Bbb R$ be defined as $f(x)= x^8 +5x^7$

Prove the function $f$ is continuous.

Proof: let $\epsilon\gt0$ be given

let $a\in \Bbb R$ be given

select $\delta \gt 0 $ such that ....$\delta=$

then for all $x \in \Bbb R$ with $|x-a|\lt\delta$

$|f(x)-f(a)|=|x^8+5x^7-a^8-5a^7|$

here where I get stuck since there are these powers, I really appreciate any hints!!

Shervan
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1 Answers1

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Hint: use Binomial Theorem for $x^{8}=((x-a)+a)^{8}$ and $x^{7}=((x-a)+a)^{7}$ and

Kavi Rama Murthy
  • 311,013