>>16654101
Hi OP

This guy: >>16654128

is correct. To elaborate a bit on that for you, continuous maps tend to mean something a little more complicated after calculus/intro real analysis. You might be used to the layman definition "Can draw without picking up pencil" or perhaps its formalization as an epsilon-delta statement. This is all true and accurate in metric spaces (which is where calculus lives), but the field of point-set topology can look at a bunch of much funkier objects.

In general, a topological space is just a set S where we specify a topology T, a special collection of subsets of S which can be combined in certain ways. We call the members of T, the "open" subsets of S. In this world, a continuous map [math]f:S_1\to S_2[/math] is just a function which maps every open set containing an input, x, into some open set containing the output, f(x). The unspoken nuance that makes this so tricky is that this definition depends on the topologies we choose for [math]S_1[/math] and [math]S_2[/math]. Sometimes authors will explicitly tell us the topology, and sometimes they'll just expect you to know it from the context (which can be a confusing pain in the ass).

Anyways, if you look at the formal definition of r(x) in this screenshot, you'll notice it has the codomain specified as [math] {-1,1} [/math]. Finite sets tend to not have many interesting/useful topologies. In a lot of use cases, the only workable topology is called the "discrete topology" where EVERY subset of S is open (i.e. in the topology). It is a simple exercise to show that any map of topological spaces where the codomain has the discrete topology is continuous by the funkier, more general definition.

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this is at the beginning of a book on algebraic topology.How is this r(x) function continuous? It flips between -1 and 1 instantly. I have not…

this is at the beginning of a book on algebraic topology.
How is this r(x) function continuous? It flips between -1 and 1 instantly.
I have not…