- 6.1: An Analytic Deﬁnition of Continuity
- Before the invention of calculus, the notion of continuity was treated intuitively if it was treated at all. At ﬁrst pass, it seems a very simple idea based solidly in our experience of the real world. Standing on the bank we see a river ﬂow past us continuously, not by tiny jerks. Even when the ﬂow might seem at ﬁrst to be discontinuous, as when it drops precipitously over a cliﬀ, a closer examination shows that it really is not. As the water approaches the cliﬀ it speeds up.
- 6.2: Sequences and Continuity
- We will examine an alternative way to prove that the function is not continuous at a≠0 by looking at the relationship between our deﬁnitions of convergence and continuity. The two ideas are actually quite closely connected, as illustrated by the following very useful theorem.
- 6.3: The Deﬁnition of the Limit of a Function
- Since these days the limit concept is generally regarded as the starting point for calculus, you might think it is a little strange that we’ve chosen to talk about continuity ﬁrst. But historically, the formal deﬁnition of a limit came after the formal deﬁnition of continuity. In some ways, the limit concept was part of a uniﬁcation of all the ideas of calculus that were studied previously and, subsequently, it became the basis for all ideas in calculus.
- 6.4: The Derivative - An Afterthought
- Along with the integral, the derivative is one of the most powerful and useful mathematical objects ever devised and we’ve been working very hard to provide a solid, rigorous foundation for it. On the other hand, now that we have built up all of the machinery we need to deﬁne and explore the concept of the derivative it will appear rather pedestrian alongside ideas like the convergence of power series, Fourier series, and the bizarre properties of Q and R .
Thumbnail: Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller. Image used with permission (Public Domain).
Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)