Chapter 4 — Further Reading

An annotated guide. You do not need to read all of it — the essential goal is to understand IVT, EVT, and the bisection method well enough to apply them confidently in every later chapter. Read by track using the guide at the end.

Mapping to the Standard Textbooks

If you are studying alongside another book, this chapter corresponds to:

Stewart, J. Calculus: Early Transcendentals (9th ed.), Section 2.5 ("Continuity").

Stewart's §2.5 covers exactly this chapter's spine: the three-condition definition, one-sided continuity, the continuity theorems for combinations and compositions, and the Intermediate Value Theorem with its root-existence corollary. Our treatment adds the classification of discontinuities into four named types, a worked bisection algorithm, the completeness-over-$\mathbb{Q}$ discussion, and the Extreme Value Theorem (which Stewart defers to §4.1). A full section-by-section crosswalk lives in appendices/appendix-h-stewart-chapter-mapping.md.

OpenStax, Calculus Volume 1 (Strang & Herman), Section 2.4 ("Continuity").

The free OpenStax treatment matches our Sections 4.2–4.3 and 4.6 closely, including removable/jump/infinite classification and IVT. It is an excellent free companion for extra worked examples and exercises; mapping in appendices/appendix-i-openstax-chapter-mapping.md.

On Continuity and the Foundations of Analysis

Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer. Chapters 4–5 give a friendly but rigorous treatment of continuity, with notably clear proofs of IVT and EVT. The single best next step if you want to see why the theorems of this chapter are true rather than merely useful. Pitched just above this textbook's "Math Major Sidebar" level.

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapters 6–7 cover continuity and the closed-interval theorems at full rigor, including the completeness-based proofs sketched in our Math Major Sidebars. Demanding but beautiful; the reference standard for math majors.

Bressoud, D. M. (2007). A Radical Approach to Real Analysis (2nd ed.). MAA. A historical route into the subject. Bressoud shows how Cauchy and Weierstrass arrived at the modern $\varepsilon$–$\delta$ definitions by wrestling with specific puzzles — exactly the puzzles (does an unbroken curve always cross a line it straddles?) that motivate Section 4.6.

On the Bisection Method and Root-Finding (Case Study 1)

Burden, R. L., and Faires, J. D. (2011). Numerical Analysis (9th ed.). Brooks/Cole. Chapter 2 is the standard textbook account of root-finding: bisection, false position, Newton's, secant, and hybrid methods, with the $\log_2$ iteration bound derived in full. Read this to see the convergence guarantee of Section 4.7 made precise.

Press, W. H., et al. (2007). Numerical Recipes (3rd ed.). Cambridge University Press. The root-finding chapter contrasts bisection with faster methods and explains the engineering wisdom behind always keeping a bracket in reserve — why production solvers like scipy.optimize.brentq combine bisection's reliability with Newton-like speed.

Heath, M. T. (2018). Scientific Computing: An Introductory Survey (3rd ed.). SIAM. Chapter 5 on nonlinear equations: compact, modern, and well-suited to readers who want the numerical-analysis viewpoint without a full course.

On the Intermediate Value Theorem and Existence Results

Körner, T. W. (2004). A Companion to Analysis. AMS. A discursive, opinionated tour of analysis that lingers on why existence theorems like IVT matter and what completeness really buys you. Good for readers who found Section 4.6's "false over $\mathbb{Q}$" argument intriguing and want more of that flavor.

Su, F. E. (1999). "Rental Harmony: Sperner's Lemma in Fair Division." American Mathematical Monthly, 106(10), 930–942. A celebrated expository paper showing how IVT-style existence arguments solve a genuine real-world division problem. A vivid example of "continuity guarantees a solution exists" beyond root-finding.

On Tax Brackets and Welfare Cliffs (Case Study 2)

Slemrod, J. (2013). Taxing Ourselves: A Citizen's Guide to the Debate over Taxes (5th ed.). MIT Press. An accessible, math-aware treatment of how tax schedules are built and why the continuity of the total-tax function $T(I)$ matters for fairness.

Saez, E. (2001). "Using elasticities to derive optimal income tax rates." Review of Economic Studies, 68(1), 205–229. The theory behind smooth optimal-tax schedules. Mathematical but readable with a calculus background; shows what economists do once they take the discontinuity-vs-continuity question seriously.

Center on Budget and Policy Priorities. https://www.cbpp.org/. Empirical documentation of welfare cliffs and of phase-out designs (like the EITC) that restore continuity to household resource functions. The policy counterpart to Case Study 2's mathematics.

On the Foundations of $\mathbb{R}$ (Math Majors)

Tao, T. (2006). Analysis I. Hindustan Book Agency / Springer. Constructs $\mathbb{R}$ rigorously from the rationals and develops continuity from there. Beautiful, careful exposition; the place to understand the completeness axiom that powers IVT and EVT.

Dedekind, R. (1872). Continuity and Irrational Numbers. (Translated reprint, many editions.) The original construction of $\mathbb{R}$ via Dedekind cuts — the historical source of the "no gaps" idea that Section 4.6 leans on. Surprisingly short and readable.

  • Physics / engineering: Burden–Faires Ch. 2 and Numerical Recipes root-finding — the practical core behind Case Study 1.
  • Economists: Slemrod and Saez, plus the CBPP material, all from Case Study 2.
  • CS / numerical analysts: Burden–Faires Ch. 2 and Heath Ch. 5; bisection is your reliability baseline.
  • Math majors: Spivak Ch. 6–7 (required), Abbott Ch. 4–5, and Tao for the foundations of $\mathbb{R}$.
  • Everyone else: Abbott Ch. 4–5 only if curious; otherwise the chapter itself plus Stewart §2.5 or OpenStax §2.4 for extra exercises is plenty.