Partial functions

partial function

Now, look at the function defined by the formula
$y = \frac{x}{2}$ .
If the input and output domains are both $ℝ$ , again this function is well-defined.
However, what if we assume that the input and output domains are both $ℕ$ ?
This time, we have a problem when $x$ is odd. For example, $\frac{3}{2}$ is not an integer, so our function is not defined for all of its input values.
It is a partial function, a concept that subsumes the notion of a function but is more general.
Chapter 2: Relations, Functions, Partial Functions, Jean Gallier.

The following applications of partial functions cannot be defined:

quot 1 0
rem 1 0
div 1 0
mod 1 0
log 0
head []
tail []

Another example is the $n$ -body problem: which has no general solution for $n$ > 2.

Due to the vagaries of the reality in which programs exist in, I/O is fundamentally partial. For reasons ranging from death to disconnection to disinterest, a user may not provide a timely and suitable response when prompted by a program!

Data obtained from other programs or input devices can also be partial. For example, a remote program or sensor may send values which cannot be processed by the receiving program - as far as the remote sensor or program is concerned, that is entirely the receiver's problem!

Ooh, mommy, mommy, what I have now doesn't work in this extremely unlikely circumstance, so I'll just throw it away and write something completely new.
Linus Torvalds

A more common solution is to let the receiver fail and be restarted.