TITLE: Am I correct in my proof that $x^2+x$ is continuous
QUESTION [5 upvotes]: I am currently trying to prove that $x^2+x$ is continuous on the reals from first principles. Let it be noted that this must be done by first principles. I am struggling to pick the correct $\delta$, but I believe I have figured it out, is this correct?
Proof:
Scratch work: $|f(a+h)-f(a)| = |((a+h)^2)+(a+h)) - (a^2 + a)| = |2ah +h^2 +h|$
Then $ |2ah +h^2 +h| = |(h(2a+1) +h^2)|$ By the triangle inequality,
$|(h(2a+1) +h^2)| \le |h(2a+1)| + |h^2|$
By definition of continuity, $|h| < \delta$ and we impose that $\delta \le 1$.
Then $|h(2a+1)| + |h^2| < |\delta(2a+1)| + \delta^2$
Because $\delta \le 1$, we know that $|\delta(2a+1)| + \delta^2 \le |\delta(2a+1)| + \delta$ and we may rewrite this as $\delta |2a+1| + \delta$
This simplifies to $\delta(|2a+1|+1)$, hence if we choose $\delta = \text{min} \{ \frac{\epsilon}{|2a+1|+1},1\}$ we will find $|f(a+h)-f(a)| < \epsilon$.
Is this correct?

REPLY [2 votes]: It should be a comment, but I make it an answer. Per your question, yes your answer is correct. Well done !