Continuity — definition, types of discontinuity (removable, jump, infinite): Visual walkthrough

Step 1 — The crayon test: what "no lifting the pen" really means

WHAT. We start with the most childlike idea: a curve is continuous if you can trace it with a crayon without lifting the crayon off the paper. Look at the two curves in the figure — the left one you can trace in one stroke, the right one forces you to lift.

WHY start here. Every symbol we write later must earn its place by encoding this one physical fact. If a formula ever disagrees with "did the crayon lift?", the formula is wrong, not the crayon.

PICTURE.

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

Notice the exact spot where the pen lifts on the right curve. That spot has a horizontal position — call it $a$ (just a name for "the x-coordinate where trouble happens"). Everything from now on is a microscope aimed at $x = a$ .

Step 2 — Two ways to "arrive" at $a$ : the one-sided limits

WHAT. To catch a lifting pen, we compare two things: the height the curve is heading toward as we slide in, versus the height it actually has at $a$ . But "heading toward" has two directions — from the left and from the right. We track them separately.

WHY separate them. Because the whole drama of a jump is that the two directions head to different heights. If we only looked at one direction we would miss half the story.

PICTURE.

Two little bugs walk toward $x = a$ :

The cyan bug approaches from the left ( $x < a$ ). The height it converges to is written $L^{-} = lim_{x \to a^{-}} f (x)$ Reading it term by term: $lim$ = "the value being approached", $x \to a^{-}$ = "as $x$ slides up to $a$ from below (the little minus means the left side)", $f (x)$ = "the height we watch".
The amber bug approaches from the right ( $x > a$ ), giving $L^{+} = lim_{x \to a^{+}} f (x)$ where $x \to a^{+}$ = "sliding down to $a$ from above".

Step 3 — The three heights that must agree

WHAT. We now have three heights at the position $a$ :

$L^{-}$ — where the left neighbours point,
$L^{+}$ — where the right neighbours point,
$f (a)$ — the crayon's actual dot.

The pen does not lift precisely when all three are the same number and that number is finite.

WHY all three. Each disagreement corresponds to a different way the pen lifts, and they can fail independently — so we truly need to check all three.

PICTURE.

The figure stacks the three heights as horizontal ticks. When they line up (left panel) the crayon glides through. When any tick is out of line (right panel) the crayon jumps.

Step 4 — Failure mode A: the hole (removable)

WHAT. Suppose the two trends agree and equal the same finite number, which we give a short name: let $L := L^{-} = L^{+}$ (so $L$ is simply "the common height both bugs report"). The neighbours all point at this height $L$ — but the dot $f (a)$ is missing, or sitting at the wrong height.

WHY it's called removable. The neighbours already agree on where the dot should go (namely $L$ ). We can remove the defect by placing (or moving) one single dot to height $L$ . One dot fixes everything.

PICTURE.

The curve sweeps smoothly toward height $L$ from both sides; there is just an open circle (empty dot) at $x = a$ , or a stray filled dot floating above.

Step 5 — Failure mode B: the step (jump)

WHAT. Now the two trends are both perfectly fine finite numbers, but they disagree: $L^{-} \neq = L^{+}$ . The left neighbours point at one height, the right at another.

WHY no single dot can fix it. A dot has one height. It can match the left trend or the right trend, never both. So no redefinition heals a jump — the gap is baked into the two different pieces.

PICTURE.

You see two curve segments landing at two different heights, with a vertical amber gap between them — the jump size $jump = ∣ L^{+} - L^{-} ∣,$ where the bars $∣ \cdot ∣$ mean "distance, always positive" (a gap has no sign, only a size).

Step 6 — Failure mode C: the pole (infinite)

WHAT. A trend fails to be a finite number at all: as a bug approaches $a$ , the height rockets to $+ \infty$ or plunges to $- \infty$ . The curve never lands. Two flavours exist:

Opposite signs — one side goes to $- \infty$ , the other to $+ \infty$ (like $\frac{1}{x - 3}$ ).
Same sign — both sides blow up the same way, e.g. both to $+ \infty$ (like $\frac{1}{x ^{2}}$ at $0$ ). Either way, at least one side is infinite, so the point is an infinite discontinuity.

WHY it's fundamentally different. In the hole and step cases the heights were real numbers we could compare. Here at least one side has no finite height — there is nothing to match to, so it is unfixable and structurally the worst. Note that even the "same sign" case is not removable: a curve heading to $+ \infty$ has no finite value a dot could take.

PICTURE.

Left panel: opposite signs (the curve dives on one side, soars on the other). Right panel: same sign (both sides soar to $+ \infty$ ), the classic $\frac{1}{x ^{2}}$ spike.

Step 7 — The subtle case: limit exists but the dot is misplaced

WHAT. This is a removable case that looks like it should be continuous. Both trends agree and are finite, the dot exists — but the dot sits at the wrong height.

WHY it still counts as discontinuous. Continuity demands trend $=$ dot. A present-but-wrong dot still breaks the equality, so the crayon still lifts (to the stray dot) then drops back.

PICTURE.

The smooth curve heads to height $1$ ; an amber dot floats up at height $5$ , disconnected.

Step 8 — Failure mode D: the blur (essential / oscillatory)

WHAT. So far every bug either settled on a height (finite) or escaped to infinity. There is one more possibility we have not shown: a bug that never settles at all — the height keeps swinging up and down forever, faster and faster, as the bug nears $a$ . Neither $L^{-}$ nor $L^{+}$ exists (not finite, not even $\pm \infty$ ). This is an essential discontinuity of the oscillatory kind.

WHY it's a fourth, separate case. Removable, jump and infinite all assumed the one-sided limits at least had answers (finite or $\pm \infty$ ). Here the very question "what height is the bug heading for?" has no answer — the trend is a blur. No dot, and no infinity, can capture it, so it is the most incurable of all.

PICTURE.

Watch the curve of $sin (1/ x)$ near $x = 0$ : the wiggles pile up infinitely tight against the $y$ -axis, sweeping the full band between $- 1$ and $+ 1$ without ever homing in on a value.

Step 9 — Edge case: continuity at a domain endpoint

WHAT. Everything above assumed $a$ has neighbours on both sides. But many functions live only on part of the line — e.g. $f (x) = x$ exists only for $x \geq 0$ , so $a = 0$ is the left edge of its domain: there is no "left side" to send a bug in from.

WHY the rule must relax. We cannot demand $L^{-}$ if there is no territory to the left — the question is meaningless there. So at an endpoint we drop the missing side and keep only the side that lives inside the domain.

PICTURE.

Only one bug (from inside the domain) approaches the endpoint; the other side is greyed out as "off the map".

The one-picture summary

One frame, five panels: glide (continuous), hole (removable), step (jump), pole (infinite), blur (essential/oscillatory). Read left to right — this is the entire classification at a glance, keyed by the same $L^{-}, L^{+}, f (a)$ ticks we built in Step 3, plus the endpoint/isolated-point reminders that interior rules need both sides while lone points are trivially continuous.

Recall Feynman retelling — the whole walk in plain words

Imagine a bug crawling along the curve toward the spot $a$ . Two bugs actually: one coming from the left, one from the right. Each bug reports the height it's heading for — those reports are $L^{-}$ and $L^{+}$ . Separately, we look at the actual dot painted at $a$ : that's $f (a)$ .

If both bug-reports and the dot are the same finite number, the crayon glides through — continuous. If the bugs agree with each other (on one height $L$ ) but the dot is missing or floating in the wrong place, it's a hole we can fill with one dot — removable. If the two bugs report different finite heights, it's a step no single dot can bridge — a jump. If either bug's report never settles — it screams off to infinity along a vertical wall — it's a pole, an infinite discontinuity (whether the two sides fly the same way, both to $+ \infty$ like $1/ x^{2}$ , or opposite ways like $1/ (x - 3)$ ). And if a bug's height just wiggles forever without ever picking a value, it's a blur — an essential, oscillatory discontinuity, the most hopeless of all.

Two twists at the edges: at the very edge of where the function lives, you only have one bug to send in — so you only ask that single side to match the dot. And at a totally isolated point with no neighbours at all, you have no bugs — so nothing can catch the crayon lifting, and the point is automatically continuous.

So: glide, hole, step, pole, or blur — those are the ways your crayon behaves, with edges and lone points handled by whichever bugs actually exist.

Recall Fast self-check

Which failure has $L^{-} = L^{+}$ finite but $f (a)$ wrong or missing? ::: Removable (hole). Which has $L^{-}, L^{+}$ finite but unequal? ::: Jump (step). Which has a side equal to $\pm \infty$ ? ::: Infinite (pole / vertical asymptote). Can both sides go to the SAME infinity and still be infinite (not removable)? ::: Yes — e.g. $1/ x^{2}$ has both sides $\to + \infty$ ; no finite dot fixes it. Which has a side that oscillates and never settles (not even to $\pm \infty$ )? ::: Essential / oscillatory (blur). Why can't a jump be patched with one dot? ::: One dot has one height; it can't equal two different trends at once. At a left domain endpoint, what must hold for continuity? ::: Only $lim_{x \to a^{+}} f (x) = f (a)$ — the missing left side is not required. Is a function continuous at an isolated point of its domain? ::: Yes — trivially (vacuously); there are no neighbours to disagree with the dot.

Connections

Continuity — definition, types of discontinuity (removable, jump, infinite) (parent)
Limits — definition and one-sided limits (the bugs are one-sided limits)
Standard limit sin(x)/x = 1 (Step 7)
Piecewise functions (Step 5)
Vertical asymptotes and rational functions (Step 6)
Differentiability implies continuity (continuity is the entry ticket for derivatives)
Intermediate Value Theorem (needs continuity across an interval, including endpoints)