Introduction to Analysis · Chapter 06

Limits and continuity of functions

The same idea as in the fourth chapter with the index replaced by a point: a tolerance answered by a distance. Then continuity, the two theorems that make a closed interval special, and the stronger condition a closed interval gives for free.

x \to a lim f (x)

∣ f (x) - L ∣ < ε

0 < ∣ x - a ∣ < δ

f (x) \to f (a)

x \to a^{+} lim f (x)

δ > 0

f \circ g

f ([a, b])

Lessons

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05One-sided and infinite limitsSome functions behave differently on the two sides of a point, and the ordinary limit then fails while each side has an answer of its own. Restricting the definition to inputs on one side gives the one-sided limits, and the ordinary limit exists exactly when both exist and agree. Replacing the band round the limit by a half-line gives limits that are infinite, and replacing the punctured interval by a half-line gives limits at infinity; in both cases only one phrase of the definition changes.Read lesson →

06Continuity at a pointContinuity at a point is the limit being what it ought to be. Three things are demanded at once: the function is defined at the point, the limit there exists, and the two agree. The definition is the limit definition with one phrase struck out — the clause excluding the point — and striking it out is exactly what forces the value to join in.Read lesson →

07Continuity on a set, and how it failsA function is continuous on a set when it is continuous at every point of it, and that is all the phrase means: the radius may be chosen afresh at each point, and usually must be. Where continuity fails, the failure has a shape, and there are only three of them — a value in the wrong place, a jump, and everything else. Naming which one has occurred is the first thing to do with a discontinuous function.Read lesson →

08Combining and composing continuous functionsAlmost every function anyone writes down is continuous, and the reason is that continuity survives every way of building one function out of others. Sums, products and quotients inherit it from the algebra of limits, and composition inherits it by a chain of two radii: the outer function names a tolerance, the inner one answers with a radius, and that radius becomes the tolerance the outermost step has to meet.Read lesson →

09The Intermediate Value TheoremA continuous function on a closed interval takes every value between the two it takes at the ends. It is the first theorem in this course that says something a picture would suggest and that fails in the rationals, and that is exactly where completeness enters: the point at which the value is taken is produced by bisection, and bisection needs nested intervals.Read lesson →

10The Extreme Value TheoremA continuous function on a closed bounded interval is bounded and attains both its bounds. Neither half is obvious and both fail if either hypothesis is dropped: on an open interval a continuous function may run away, and on a closed unbounded one it may approach a level it never reaches. The proof is Bolzano–Weierstrass used twice, and it is the theorem that makes the search for a maximum a sensible thing to do at all.Read lesson →

11Uniform continuityContinuity lets the radius be chosen afresh at each point. Uniform continuity demands one radius that serves the whole domain at once. The difference is a swap of two quantifiers, and it is a real difference: some continuous functions need ever smaller radii as they move about their domain, and have no single one. The theorem of this lesson is that on a closed bounded interval the difficulty cannot arise.Read lesson →

12Chapter practiceThis lesson mixes the chapter without announcing which idea each question is about. Deciding whether a question is about a limit, about continuity at a point, about one of the two theorems on a closed interval, or about the quantifier order, is part of the work.Read lesson →

Mapping the lesson.

Limits and continuity of functions

Build the foundation