I do not see anything hideous with this definition, it is just the right definition.

Often, the function domain is not defined (strictly speaking, this is a logical error, of which I am guilty as charged), it is assumed that it is the maximum subset of some other set (identity of which is obvious from the context) for which the rule makes sense. For example, when working with sequences (i.e., functions that assign real numbers, complex numbers, vectors, etc. to non-negative integers n ³ 0), the function domain is usually all n ³ 0 and the rule is how you describe the sequence, for example

a_n = (1 + 1/n)ⁿ

Some rules do not make sense for all n ³ 0, usually when you get a zero in the denominator. The above rule does not make sense if n = 0 and the maximum domain is n > 0 (or n ³ 1, whatever Description you choose). It is also clear that the rule never assigns 2 different real numbers to a particular integer k > 0. In elementary calculus, we work with real functions of one real variable, i.e., with rules that assign to every real number from the domain some other real number. The function domain is often all real numbers and the rule is how you describe the function, for example

f(x) = x²·sin(1/x)

The above rule does not make sense if x = 0 and therefore the maximum domain implied by the above rule is x ¹ 0. However, it is not always useful to limit yourself to the maximum domain for which the rule makes sense - you can miss interesting things. For example, I can define the above function for all x as follows (strictly speaking, the domain is different and therefore it is a different function):

g(x) = x²·sin(1/x) for x ¹ 0
g(0) = 0

When you try to draw a graph of this function, it oscillates faster and faster as x ® 0, so you never really get there. However, it is easy to show not only that the function g is continuous at x = 0, it even has a derivative at this point.

Real functions of one real variable (i.e., functions investigated in elementary calculus) may have various properties - they can be continuous, periodic, etc. Then again, they may not have some or all of the above properties. So it would not really make sense to include those properties in the definition of a function, the definition should include as many useful functions as possible. With elementary functions, properties like continuity or periodicity may seem obvious, but generally, this is not always true. Consider a periodic function with a period p > 0:

f(x + p) = f(x)

If p is the period, then surely 2p, 3p, ... are also periods, but they are all greater than p. May there be a smaller period than p? If yes, what is it? Once I saw the following theorem (as the 1st theorem in a book about Fourier series'):

If a non-constatnt periodic function f is continuous in at least one point, then the function has a minimum period p_min > 0.

The proof is really simple - by assuming that the minimum period does not exist, it is easy to show that the function f must be a constant. Now apply this theorem to Dirichlet's function:

D(x) = 0 if x is irrational
D(x) = 1/q if x is rational and can be reduced to x = p/q

The function is continuous for every irrational x and discontinuous for every rational x. It obviously has period 1, maybe less. The theorem says that Dirichlet's function has a minimum period. What is it?

I guess the best example that I can come up with is pressure. In still water, pressure is a function of depth. This means that as I go deeper into the ocean, the water pressure will get bigger.

I usually think of acceleration as a function of time, because the longer one accelerates, the faster one goes.

I could probably think of more real world examples (e.g. a person's grade in math is a function of their intelligence, effort, and previous training) but the above examples are the ones I usually use to explain functions to people who aren't receptive to the confusing language of mathematical definitions.

--CS