Treat a function as a rule from real inputs to real outputs, then read its domain, range, graph, and level sets. You will also build simple examples from formulas, cases, and restrictions.

Practice direct proof, contradiction, contrapositive proof, and proof by example or counterexample. These patterns become the daily tools for showing that a function really has a surprising property.

Compare finite, countable, and uncountable sets, including the rationals and reals. This explains why some functions can be controlled by listing points while others cannot.

Use sequences to capture “getting close” and test limits through concrete estimates. This gives a first precise way to talk about behavior near a point.

Use least upper bounds, greatest lower bounds, nested intervals, and completeness of the real numbers. Many pathological examples rely on what the real line contains that the rationals do not.

Work with neighborhoods, open sets, closed sets, accumulation points, and dense sets. These ideas explain how a set can be spread everywhere while still being small in another sense.

State and prove continuity using sequences and epsilon-delta estimates. You will see how small input changes are supposed to force small output changes.

Build functions using piecewise rules, indicator functions, max and min, sums, products, and compositions. These construction methods are the workshop for almost every example later in the course.

Trace how nineteenth- and twentieth-century analysts created counterexamples that changed calculus, integration, and set theory. Weierstrass, Cantor, Baire, Lebesgue, Volterra, and Banach show why pathological functions became central rather than decorative.

Classify removable, jump, and infinite discontinuities through limits from one side and two sides. You will make functions with exactly chosen failures of continuity at selected points.

Use the density of rationals and irrationals to build the Dirichlet function and related everywhere-discontinuous examples. This chapter shows how two interwoven sets can force total instability.

Study Thomae’s function and variants that are continuous at every irrational and discontinuous at every rational. You will see how countability can make a dense set of bad points thin enough to control.

Measure local wildness with oscillation and use it to describe where a function is continuous. This gives a practical test for proving that a chosen set is exactly the set of discontinuities.

Use compactness, the extreme value theorem, and uniform continuity on closed intervals. These results mark the boundary between functions that can be wild and functions that are forced to behave.

Distinguish pointwise convergence from uniform convergence for sequences of functions. You will build limits of functions while tracking which properties survive the limiting process.

Create functions that appear as pointwise limits of continuous functions and see why new discontinuities can appear. This leads to Baire class one functions and the rule that their continuity points cannot be arbitrary.

Work with Borel sets and Baire classes as a hierarchy of functions built from open sets and limits. This gives a map for locating many pathological functions without treating them as random tricks.

Define derivatives from difference quotients and prove the standard rules. You will connect slopes, tangent behavior, and local linear approximation before studying how derivatives can misbehave.

Study derivatives that are discontinuous, unbounded, or hard to recognize, while still obeying Darboux’s intermediate value property. Examples from Volterra and related constructions show that derivatives are not just arbitrary functions.

Build and analyze continuous functions with no derivative at any point, including Weierstrass-type and Takagi-type examples. The chapter connects jagged geometry, infinite series, and careful estimates.

Use total variation to separate smooth-looking functions from functions that wiggle too much. Monotone functions and bounded-variation functions become a bridge between calculus and measure theory.

Define Riemann sums, integrability, and the role of discontinuities in Riemann integration. You will test classic pathological functions to see which ones have areas and which ones do not.

Construct the Cantor set step by step and prove that it is closed, uncountable, nowhere dense, and built from removed intervals. It becomes a reusable machine for making functions with surprising size and shape.

Assign length to sets with Lebesgue measure and distinguish null sets from large uncountable sets. This lets you describe functions that are bad on many points but harmless for integration.

Use measurable functions, simple functions, and key convergence theorems to handle limits under the integral sign. These tools make many wild functions manageable instead of mysterious.

Compare absolute continuity, differentiability almost everywhere, and the fundamental theorem of calculus. You will see exactly when derivatives and integrals undo each other and when they do not.

Analyze the Cantor function: continuous, increasing, constant on many intervals, and with derivative zero almost everywhere. It shows how a function can climb from 0 to 1 while hiding nearly all of its increase.

Work with functions that are considered equal almost everywhere and measure their size with Lp norms. This chapter explains how analysts study functions by ignoring changes on null sets.

Use choice principles to build Vitali sets and functions that cannot be Lebesgue measurable. These examples show where ordinary length and integration truly break.

Use vector spaces over the rationals and Hamel bases to solve Cauchy’s equation in discontinuous ways. You will see why mild regularity assumptions are needed for familiar algebraic laws to force linear functions.

Use nowhere dense sets, meager sets, comeager sets, and the Baire category theorem. Category gives a second meaning of “small” and “large,” different from measure.

Apply category ideas to function spaces with the uniform metric. You will prove why “typical” continuous functions can be nowhere differentiable, even though smooth functions are easier to draw.

Study trigonometric series and Fourier series that converge badly, diverge at points, or represent functions in unexpected ways. This connects pathological real functions to approximation, signals, and harmonic analysis.

Follow a full construction from a requested property list to a finished example: choose the right small set, define the function, prove each property, and check for hidden contradictions. This is the main workflow used in real counterexample building.

Use plotting, numerical sampling, symbolic tools, and formal proof assistants without trusting pictures too much. You will see how modern tools can suggest examples, expose mistakes, and support rigorous proof writing.

Connect the course to counterexample books, real analysis texts, journals, seminars, and open formalization libraries. The chapter covers proof portfolios, topics for further study, and paths into analysis, topology, measure theory, and mathematical research.