Alternatively, defining f(0) to be infinity (a projectively extended real number) also allows f to be continuous.

Another possibility is to define f(0) to be some kind of "bottom" value. See the following tweets by Andrej Bauer: https://twitter.com/andrejbauer/status/1268860981943439361

Continuity is important because without it we lose computability. Every computable function is continuous (in an appropriate sense).

But, I don't think what you said is quite right, either. If you define f(x) = 1/x over the non-zero reals, then the range does have the same type as the domain- you can't get zero out of the function, so the range is also the non-zero reals.

Not quite. The extended reals are ℝ + {inf, -inf}. This has the problem that 1/x approaches inf from the right and -inf from the left.

As ogogmad notes, in order to define a value for 1/x at 0, you need the projectively extended reals.

You are both right. There are several, different, extensions of the reals. The "projective" extension turns the real line into a (topological) circumference by adding a single point at infinity. The "affine" extension turns it into a closed segment by adding two infinities, one at each end; this corresponds to the convention used by ieee floating point numbers. There are many more extensions, like complex numbers, dual numbers, p-adic numbers, and in many of them (except complex numbers) you have various kinds of divide-by-zero fun.

For example, if we take a mercator projection of the globe onto a 2-infinite-ended cylinder, the latitude coordinate could meaningfully be either +∞ or –∞. If we take the stereographic projection of the globe onto the plane, every pair of coordinates with one infinite value corresponds to the same point on the sphere.