Lecture 13: Achilles, Tortoises, Limits and Continuity

The lecture begins with an introduction to the fundamental concept of calculus, the limit, which is essential for defining both the derivative and the integral. The instructor acknowledges the vagueness in the historical definition of limits and the challenge it posed to formalize mathematically. The goal of the lecture is to define the limit rigorously and discuss foundational ideas of calculus, including continuity and differentiability. The historical context is provided by referencing Newton's and Leibniz's understanding of limits, which was intuitive but not mathematically rigorous until the 19th century.

The second paragraph delves into Zeno's Paradox of Achilles and the Tortoise to illustrate the concept of limits. It describes a race where Achilles, being faster, gives the tortoise a head start. The paradox suggests that Achilles can never overtake the tortoise because for every point Achilles reaches, the tortoise moves slightly ahead. The lecture uses this paradox to introduce the idea of real numbers and the limit, explaining that Achilles and the tortoise will eventually meet at a specific point, represented by the repeating decimal 1.1111..., which is equal to 1.

This paragraph explores the concept of real numbers and their relation to limiting processes. It presents the idea that a decimal point followed by an infinite string of digits defines a real number. The example of 0.999999... is used to demonstrate that this repeating decimal is equal to 1, not just almost 1. The paragraph also discusses the application of limits to derivatives, showing how the limit process is used to define the derivative of a function, such as f(x) = x^2 at the point x = 1.

The fourth paragraph provides a formal definition of a limit in the context of a function. It explains that a limit of a function f(x) as x approaches a value C is a number L, which is approached by the function's values as x gets arbitrarily close to C. The formal definition is presented using the Greek letters epsilon (ε) and delta (δ) to describe the relationship between the challenge distance around the limit and the interval around C. The paragraph also includes a graphical explanation and a historical note on when this definition was established.

The lecture continues with a formal proof of a limit, using the example of the function f(x) = 2(x^2 - 4)/(x - 2) and demonstrating that the limit as x approaches 2 is 8. The proof involves choosing a delta value in response to a given epsilon challenge, showing that the function values are within the epsilon neighborhood of the limit. The concept of continuity is introduced as a function whose value at a point is predictable from its neighboring values, and it is formalized as the limit of the function values as x approaches a point being equal to the function's value at that point.

The sixth paragraph discusses the concept of continuity, using the analogy of a car driving on a road to explain that a continuous function's value at any point can be predicted by its neighboring values. It contrasts continuous functions with those that have jumps or kinks, which are not continuous. The paragraph then introduces differentiability, explaining that a function has a derivative at a point if it can be magnified to look like a straight line. Functions with kinks or sharp points are not differentiable, whereas differentiable functions must also be continuous.

The final paragraph explores differentiability in the context of real-world functions. It notes that while some functions, like those with kinks, are not differentiable, many functions used to describe the physical world are differentiable. Examples include polynomials, trigonometric, exponential, and logarithmic functions, all of which can be combined in various ways to form new differentiable functions. The paragraph also mentions that while functions with a kink at a single point are differentiable everywhere else, it is possible to construct a function that is continuous but nowhere differentiable, characterized by an extremely jagged graph.