Foundations — Continuity — definition, types of discontinuity (removable, jump, infinite)

Before you can read the parent note Continuity, you must own every symbol it throws at you. Below, each symbol gets three things: plain words, the picture, and why the topic needs it. They are ordered so each one leans only on the ones above it.

1. What a function $f$ and $f(a)$ mean

The picture. Draw a horizontal axis for the input $x$ and a vertical axis for the output. For each input $x$ you place a dot at height $f(x)$. Sweep $x$ across and the dots trace the graph. The single dot directly above $x=a$, at height $f(a)$, is "the value the function actually takes at $a$."

Why the topic needs it. Continuity's condition 1 is literally "$f(a)$ is defined." If the machine chokes on the input $a$ — like dividing by zero — there is no dot above $a$, and the pen-stroke has a gap before we even ask about approaching.

2. The arrow $x\to a$ and the idea of "approaching"

The picture. Put a marker on the input axis and let it crawl toward $a$, never landing on it. Watch what happens to the height of the dot as the marker crawls. The arrow $\to$ is that crawling motion.

Why the topic needs it. Continuity compares two different things: where the graph is heading versus where it lands. The arrow captures "heading" without ever touching the landing spot — that's the whole trick that lets a hole exist at $a$ while the graph still heads somewhere sensible.

3. The limit ${\lim }_{x\to a}f(x)$

The picture. As your marker slides toward $a$, the dot's height presses toward one horizontal level $L$. Even if there is a hole exactly at $a$ (no dot there), the height it was pressing toward is still $L$. The limit reads that "target height," ignoring the point itself.

Why this tool and not just $f(a)$? Because $f(a)$ can be missing, wrong, or in the wrong place, while the trend of the surrounding graph still points clearly at a value. The limit is the tool that measures the trend independently of the single point — exactly what we need to detect holes and jumps.

For a deeper build of this idea see Limits — definition and one-sided limits.

4. One-sided limits $L^{-}$ and $L^{+}$

The picture. Two separate marchers approach the vertical line $x=a$: one from the left, one from the right. Each reports the height its side is pressing toward. If both report the same height, the two-sided limit exists and equals that height. If they disagree, there is no single two-sided limit.

Why the topic needs it. The jump discontinuity is defined entirely by $L^{-}\ne L^{+}$: the two marchers land at different heights, a stair-step. Without splitting the approach into two sides, you literally cannot describe a jump. See Piecewise functions for where these two-sided splits come from.

5. Infinity $\pm \infty$ and blowing up

The picture. Near $x=a$ the graph hugs a vertical line and rockets upward (or downward) alongside it. That vertical line the graph never crosses is a vertical asymptote.

Why the topic needs it. The infinite discontinuity is exactly this rocket. And note the signs can differ on each side: for $\frac{1}{x-3}$, a tiny negative denominator just below $3$ gives $-\infty$, a tiny positive denominator just above gives $+\infty$. Handling both signs is why the parent checks each side separately. See Vertical asymptotes and rational functions.

6. The $\frac{0}{0}$ symptom and cancellation

The picture. Both numerator and denominator curves cross zero at the same input $a$. At $a$ itself the fraction is meaningless (no dot), but just beside $a$ the ratio may settle onto a clean finite number — that settled number is the limit.

Why the topic needs it. Removable discontinuities almost always show up as $\frac{0}{0}$. Factoring and cancelling a common $(x-a)$ factor reveals the height the graph was heading toward. That's precisely why the parent's Example 1 factors $\frac{x^2-1}{x-1}=x+1$ (for $x\ne 1$) to expose the limit $2$.

7. The standard limit $\frac{\sin x}{x}\to 1$

Plain words. As the angle $x$ (in radians) shrinks toward $0$, the ratio of $\sin x$ to $x$ presses toward $1$ — even though at $x=0$ the fraction is the warning-light $\frac{0}{0}$.

Why the topic needs it. The parent's Example 4 uses it to say the limit exists and equals $1$, so a function that sets $g(0)=5$ has a removable flaw (limit exists, value misplaced). The full geometric build lives at Standard limit sin(x)/x = 1 — here you only need to trust the value $1$.

How the foundations feed the topic

Read it top-down: the function and the approach-arrow build the limit; the limit splits into two sides; agreement or disagreement of the sides, plus the value $f(a)$, decides continuity or the three flavors of failure.

Equipment checklist

Test yourself — you're ready for the parent note only if each reveal is obvious.

Connections

• Continuity — definition, types of discontinuity (removable, jump, infinite) (this page feeds it)
• Limits — definition and one-sided limits (the machinery of $\lim$, $L^{-}$, $L^{+}$)
• Standard limit sin(x)/x = 1 (the imported fact in §7)
• Piecewise functions (source of two-sided splits and jumps)
• Vertical asymptotes and rational functions (source of $\pm \infty$ blow-ups)