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Functions

The value at $𝑎$ is written $𝑓(𝑎)$. Two functions $𝑓:𝐴\to 𝐵$ and $𝑔:𝐶\to 𝐷$ are the same when $𝐴=𝐶$, $𝐵=𝐷$, and $𝑓(𝑎)=𝑔(𝑎)$ for every $𝑎\in 𝐴$ — even if the expressions look different. See Spivak (1967).

The graph is a set of pairs, not the rule:

A picture of that set in the plane is useful, but it is not the function. The function is the rule. I keep wanting to say “the function is the curve”; then I remember that two different-looking formulas can draw the same curve, and that some functions have no curve I can draw at all.

A first family

The identity $𝐼:\mathbb{R}\to \mathbb{R}$ given by $𝐼(𝑥)=𝑥$ is the dullest interesting example. A constant function ${𝑐}_{𝑏}:\mathbb{R}\to \mathbb{R}$ given by ${𝑐}_{𝑏}(𝑥)=𝑏$ forgets its input entirely. Absolute value is the first one that is not a single algebraic expression:

Each piece is a function on a smaller domain; glued together they still assign one output to each real $𝑥$. That is all I am asking for. At $0$ the two clauses agree, so there is no fight about the value.

def abs(x):
    return x if x >= 0 else -x

The picture is Figure 1.1, the table is Table 1.1, and the listing is Listing 1.1. All three are the same function (1.2), written three ways.

Domain is part of the data

The same formula on two domains is two functions. Restriction of $𝑥\mapsto |𝑥|$ to $[0,\infty )$ is the identity on that interval; restriction to $(-\infty ,0]$ is $𝑥\mapsto -𝑥$. I will need that distinction when I ask for an inverse.

I am not requiring a closed-form expression, and I am not requiring that the graph be drawable in one stroke. Continuity comes later. For now, a function may jump, or exist only on $(0,1)$, or be defined by a different clause on each integer. The definition does not care.