Foundations — Functions, limits, and continuity

Before you can read the parent note, you need to read its symbols. Below is every notation, letter, and idea the parent leans on, in the order that each one depends only on the ones before it. A smart 12-year-old who has never seen any of this should be able to start at line one.

1. Sets and the "is inside" symbol $\in$

WHAT it looks like: a boundary drawn around some dots. A dot inside the boundary "belongs"; a dot outside does not.

WHY the topic needs it. A function has to know which inputs it is allowed to eat. That allowed collection is a set. Without "is inside", we could not even state "for each input $x$ inside the domain".

2. Numbers on a line, and the real numbers $\mathbb{R}$

WHAT it looks like: one straight line, no gaps, every spot filled — whole numbers, fractions, and endless-decimal numbers like $\sqrt{2}$ all live somewhere on it.

WHY the topic needs it. Limits are all about sliding along this ruler toward a spot. The word "creeps toward" only makes sense because the ruler has no gaps between points.

3. Inputs, outputs, and the arrow $\to$

WHAT it looks like: a box labelled $f$. An arrow goes in carrying $x$; a different arrow comes out carrying $f(x)$.

WHY "exactly one" matters (the picture). Look at the machine: from any single input dot, exactly one arrow leaves. If two arrows left the same input, you couldn't say what the machine "does" — that ambiguity is exactly what the vertical line test catches on a graph.

Range-vs-codomain :: The codomain is the shelf you might reach; the range is the cans you actually grabbed.

4. The graph and the two axes

WHY the topic needs it. "No jumps, holes, or explosions" and "trace without lifting the pen" are statements about the picture. The vertical line test lives here too.

5. Absolute value $|\cdot |$ — the "distance" bars

WHY this tool and not just subtraction? Subtraction $x-c$ can be negative (if $x$ is left of $c$) or positive (right of $c$). But when we ask "how close are they?", left and right shouldn't matter — only the gap. The bars throw away the sign so both sides count equally. This is the exact tool the limit needs to say "approach from either side".

6. The Greek letters $\epsilon$, $\delta$, $\theta$

WHY these names. They're just labels, but the convention is universal: $\epsilon$ and $\delta$ always mean "small positive amounts" in limit-land. Reading $\epsilon$–$\delta$ is easier once you hear "output-tolerance / input-closeness" instead of "epsilon / delta".

7. The limit symbol $\underset{x\to c}{\lim }$

WHY a new symbol at all? We already have $f(c)$ — the value at $c$. The limit is a different question: not "what is the output at $c$?" but "what is the output aiming for near $c$?" Those can disagree (a hole, a jump). We need a separate symbol because they are separate ideas — this is the whole "Approach ≠ Arrive" point of the parent note.

8. The $\pm \infty$ symbol and "blow up"

WHY the topic needs it. One kind of broken behaviour is an infinite discontinuity (like $1/x$ near $0$): the output explodes. We need a word for "explodes" so we can name and exclude it.

9. Putting the alphabet together — $0/0$ and indeterminate forms

WHY it appears. When both top and bottom of a fraction head to $0$ at the same spot (like $\frac{x^2-1}{x-1}$ at $x=1$), plugging in gives $\frac{0}{0}$. The race between top and bottom decides the true limit — you settle it by factoring, and later by L'Hôpital's Rule.

Prerequisite map

Equipment checklist