The epsilon-delta-criterion aims to formalize our intuitive understanding what it takes for (a part) of some function to be continuous, i.e, replacing the „idealized pencil“ by requiring that for some sufficiently small variation of the argument, the changes in value become arbitrarily small. As often is the case with formalisms, this gives raise to wild examples. For example, https://en.m.wikipedia.org/wiki/Thomae%27s_function is continuous almost everywhere, yet the points of discontinuity lie dense. One might reject the idea of continuity in a single point as nonsensical. Playing the devils advocate, one should only be allowed to assign this property to some section of the function graph (if my pencil just marks a point, then what is the point). I am not very literate in the history if analysis. Were there alternative proposals put forward that would judge Thomae‘s function differently?

The epsilon-delta formulation of continuity arrived at a time when the concept of function itself was not entirely definite. Dedekind had his own approach to continuity, involving something like his cuts, but on the graph of the function, and of course there were also infinitesimal approaches to continuity---an infinitesimal change in x causes at most an infinitesimal change in f(x). I think the mathematical community took quite some time coming to a stable view of what it means to be a function and what it means for a function to be continuous.

What do you think of the definition of continuity in terms of open balls? I find that version somewhat easier to think about than the epsilon-delta version. I learned of it in D. J. Bernstein’s 1997 essay “Calculus for Mathematicians”:

> Definition 2.1. Let 𝑓 be a function defined at 𝑐. Then 𝑓 is continuous at 𝑐 if, for any open ball 𝐹 around 𝑓(𝑐), there is an open ball 𝐵 around 𝑐 such that 𝑓(𝐵) ⊆ 𝐹.

I view the open-ball definition as expressing exactly the same concept as the epsilon-delta formulation, since what a ball is, is the set of points within some positive distance of the given point. Meanwhile, to be sure, the open ball way of thinking about this leads eventually to the core ideas of topology, where one finds vast generalizations of continuity to all kinds of other topological spaces.

The epsilon-delta-criterion aims to formalize our intuitive understanding what it takes for (a part) of some function to be continuous, i.e, replacing the „idealized pencil“ by requiring that for some sufficiently small variation of the argument, the changes in value become arbitrarily small. As often is the case with formalisms, this gives raise to wild examples. For example, https://en.m.wikipedia.org/wiki/Thomae%27s_function is continuous almost everywhere, yet the points of discontinuity lie dense. One might reject the idea of continuity in a single point as nonsensical. Playing the devils advocate, one should only be allowed to assign this property to some section of the function graph (if my pencil just marks a point, then what is the point). I am not very literate in the history if analysis. Were there alternative proposals put forward that would judge Thomae‘s function differently?

The epsilon-delta formulation of continuity arrived at a time when the concept of function itself was not entirely definite. Dedekind had his own approach to continuity, involving something like his cuts, but on the graph of the function, and of course there were also infinitesimal approaches to continuity---an infinitesimal change in x causes at most an infinitesimal change in f(x). I think the mathematical community took quite some time coming to a stable view of what it means to be a function and what it means for a function to be continuous.

What do you think of the definition of continuity in terms of open balls? I find that version somewhat easier to think about than the epsilon-delta version. I learned of it in D. J. Bernstein’s 1997 essay “Calculus for Mathematicians”:

> Definition 2.1. Let 𝑓 be a function defined at 𝑐. Then 𝑓 is continuous at 𝑐 if, for any open ball 𝐹 around 𝑓(𝑐), there is an open ball 𝐵 around 𝑐 such that 𝑓(𝐵) ⊆ 𝐹.

I view the open-ball definition as expressing exactly the same concept as the epsilon-delta formulation, since what a ball is, is the set of points within some positive distance of the given point. Meanwhile, to be sure, the open ball way of thinking about this leads eventually to the core ideas of topology, where one finds vast generalizations of continuity to all kinds of other topological spaces.