If this was true:

tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1

Wouldnt it imply that

tan(π/8) would be halfway between 0 and 1 ie .5?

By my calculations it is 0.414 or √2 - 1

ALso with:

• f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1.

wouldnt this mean that f(0.25) is supposed to be half way between 0 and 1 or .5. However f(0.25) = 0.25/0.75 = 1/3

It implies that tan(π/8) would be "halfway" between 0 and 1 in this sense of what "halfway" means. In this sense, 0.5 is not halfway between 0 and 1, 0.414 is.

It's easier to visualize it. You're standing on the roof of a 1-meter-tall building on a flat earth with a perfectly clear atmosphere. If you look straight out (90°) you can see to infinity (tan 90°). If you look down (0°) you can see where you are (tan 0°). If you look halfway between those (45°), you can see 1 (tan 45°) meter straight in front of you. But if you look halfway down again, you won't see exactly 0.5 meters, will you?

The original issue at hand is to talk sensibly about “half way” between 0 and infinity. But in the standard way we think about distance between numbers, there’s obviously no way to do that — you can’t add and subtract real numbers to infinity!

So, implicitly what’s happening here is that by talking about a map between a finite interval and (0,inf), we are equipping (0,inf) with a new, special definition of distance between numbers. This is called a metric.

Usually, when we talk about the distance between x and y, we mean `d(x,y) = |x-y|` — this is called the Euclidean metric (in 1 dimension).

Here, we’ve introduced a new metric on (0,inf): `d2(x,y) = |tan^-1(x) - tan^-1(y)|/(pi/8)`

The half way point between 0 and 1 under the Euclidean metric is 0.5. The half way point between 0 and 1 under our fancy new metric d2 is ~0.414.

TL;DR: You can generalize the notion of “distance between two numbers”, using distance functions called metrics. Under the typical metric, the half way point between 0 and 1 is 0.5. Under our cool new infinite-tan metric, the half way point between 0 and 1 is ~0.414.

tan(π/8) = √2 - 1, not 1/2, but the value is indeed halfway between 0 and 1 if you take the distance function to be

d(a, b) = |b - a| / (√(1 + a²)√(1 + b²)) or

d(a, b) = |b - a| / |1 + ab|

(These are chordal distance or stereographic distance, respectively, when the real number line is used to represent points on a circle under stereographic projection.)

The halfway point is 0.5 when you use the distance function d(a, b) = |b - a|.