Functions are vectors

> A function f from a set X to a set Y, denoted f: X -> Y, is a relation from X, the domain, to Y, the co-domain, that satisfies two properties: (1) every element in X is related to some element in Y, and (2) no element in X is related to more than one element in Y.

Susanna S. Epp. 2010. Discrete Mathematics with Applications (4th. ed.). p.384

1. When we want to consider both positive and negative square roots, we can just say that instead of the function sqrt(x): ℝ ⟶ ℝ which always gives a nonnegative real number, we are using a function full_sqrt(x): ℝ ⟶ ℝ² which gives an ordered pair of real numbers.

2. We mostly define arcsine as yielding a single value between -pi/2 and pi/2, but it's just as easy to define the equivalence relation "x ~ y when sin(x) = sin(y)" and then say that the unique value given by the arcsine function is one of the equivalence classes which that relation induces over the real numbers -- or that it is the subset of the reals included in such an equivalence class.

3. If we are concerned about graphing a multi-valued function, we can do that too; for your parabola, we can define the three-dimensional function "z(x,y): ℝ² ⟶ {0, 1} = 1 when x = y² and 0 otherwise" and the graph of this function is the parabola you want.

So you're wrong about the definition of a function, except that you're right in spirit; the way we normally think about functions, as opposed to the way we define them, definitely allows them to yield multiple values for one input.

You're catastrophically wrong about bijective functions; those are an important concept and your definition is not related to the actual one. A bijective function is one for which all values in the target set are reached by exactly one, no more and no less, value in the source set.

The significance of this paradigm shift is that it allowed mathematicians to apply some of the geometric intuition developed from the study of finite-dimensional spaces (such as the 3D Euclidean space) to difficult questions involving functions, such as the existence of solutions to certain differential equations.

The history of this change of perspective is absolutely fascinating and can be traced back to the end of the 19th century and beginning of the 20th century. At the time, work on axiomatic foundations of mathematics was driving a systematization of the study of mathematical objects by capturing their structure with a concise list of axioms. This is for example how the concept of an abstract vector space was born, encompassing not only Euclidean spaces but also infinite-dimensional spaces of functions.

An early reference already demonstrating this change of perspective, albeit in a primitive form, is a memoir by Vito Volterra from 1889 [1]. The PhD thesis of Maurice Fréchet from 1906 [2] is arguably the work that was most influential in crystalizing the new paradigm and presenting it in a modern form that served as a key reference for the first half of the 19th century. Of course, these are only two among a multitude of works around that time. Looking at later developments in the 19th century, it is hard not to also mention the book by Stefan Banach from 1932 [3].

[1] https://projecteuclid.org/journals/acta-mathematica/volume-1...

[2] https://zenodo.org/record/1428464/files/article.pdf

[3] http://kielich.amu.edu.pl/Stefan_Banach/pdf/teoria-operacji-...

Any claims of injective or surjective properties (and by extension bijection) are moot without the domain and co-domain being specified.

To map math ↦ dev semantics: functions need to be pure, deterministic and declared with strong types. The properties of square root in particular will vary wildly based on what those types are.