What would change if the definition of a function were the definition of a relation? So, I was confused by this since the first time I have first learned about functions: why do we require, that every element in the domain maps to one and only one element in the range. (Is that even true for a more  rigorous formulation?)
To give it more accuracy: what results would change, if we didn't require that condition and allow a function to be any kind of relation? 
 A: I elements could map to several elements in the range, the function value wouldn't be well-defined. If both $(0, 2) \in f$ and $(0, 5) \in f$ then what would be the value of $f(0) + 3$?
However, in complex analysis one sometimes uses multivalued functions.
A: As lulu commented, a function is a relation, but a stricter one. 
Usually any elements can be the subject of a relation. Take the addition as an example: you can add any to numbers: $+(a,b)$, thus there can exist such a binary relation between any two numbers. However, a function marks a "relation" between two "special" (or "exact") numbers. For example $f:\mathbb{C}\rightarrow\mathbb{C}, \ f(x) = 2x$ cannot describe a relation between $3$ and $4$, since $f(3)=6$.
The reason why we are using functions is that they help us model certain real - life problems. Consider for example, the complexity of a program (Computer Science), which is evaluated through a function which gives us the number of steps that an algorithm requires to finish on a given data size. Namely, the quicksort algorithm maps an $n$-size array (list of elements) into $nlog_{2}{n}$ number of steps.    
