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By cluttered $\epsilon$, I refer to $C\epsilon$ "where $C$ is the constant $2/|b| + |2a/b^2|$" on p. 18 beneath. See the green arrows.

By unkempt $\delta$, I mean $\delta$'s like $\min(4\varepsilon -\varepsilon^2, 4\varepsilon + \varepsilon^2)$, $\log_2(\epsilon+4)-2$, $\min\left[\dfrac1{2}, \dfrac{\epsilon}{e^{1.5}}\right]$, $\min\left(\dfrac1{2},\dfrac{\epsilon}{\dfrac{1}{2}+2|a|}\right)$, $\min\left(1,\dfrac{\epsilon}{2(|a|^2+1)(2|a|+1)}\right)$.

  1. Most textbooks conclude $\delta-\epsilon$ proofs tidily with $\epsilon > 0$ alone, in red beneath. But what's the official term for this alternative $\delta-\epsilon$ proof that concludes $\delta-\epsilon$ proofs with a cluttered $\epsilon$, in green beneath?

  2. I forgot the particulars of another textbook (not the one beneath) that recommended the green alternative, because it's quicker to define a new $\epsilon_2 := \text{ cluttered } \epsilon_1$ at the end! Don't work backwards to deduce unkempt $\delta$'s. Please recommend more such textbooks?

Frank Morgan, Real Analysis (2005), pages 17-8.

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