# What is this commutative diagram for continuity?

The "Homological cohomology memes" facebook page recently posted an image, copied below.

In the 4th panel, there are maps from $c$ into $\mathbb{R}$, from $\ell_\infty$ into $\mathbb{R}$, $f: \mathbb{R} \to \mathbb{R}$, and $A_f: c \to \ell_\infty$. What are these maps?

Panel 1: $\epsilon$-$\delta$ continuity

Panel 2: sequential continuity

Panel 3: topological continuity

Panel 4: ???

Panel 5: continuous functor

• might this just be the definition in terms of sequences? $f$ continuous if for all convergent sequences $(x_n)_{n\in \mathbb N}$ (with $x_n \rightarrow x$), we have $f(x_n)\rightarrow f(x)$? – slin0 Feb 17 '17 at 10:11
• I think @slin0 is right, but this definition is very flawed, since it is a priori not true that $A_f$ (apply $f$ to the sequence) takes a convergent sequence to a bounded sequence. – MooS Feb 17 '17 at 10:18
• @MooS: sure it is 'a priori true', just you haven't proven it yet, but it's just as much of a problem to discuss c (i.e. the space of convergent sequences) without characterizing continuity, either. the original paper takes the conventional definition for granted and then shows their equivalence, fyi – oldrinb Feb 26 '17 at 2:36
• @oldrinb That paper you linked answers the question, I think. Would you like to write an answer? – 6005 Feb 27 '17 at 23:15
• This was posted by @Trivbycategorytheory as an answer because he had not enough reputation to post a comment: Only posting for context. For panel 4, the thought was simply "continuity through commutative diagram". Indeed, the definition we used was that found in the link posted by oldrinb. I'm not sure if it is a standard definition as I haven't encountered it. – miracle173 May 4 '17 at 15:07