Here’s the intuition before any rules. A continuous function is one you can trace in a single unbroken stroke. The moment you have to lift your pen — to skip a hole, leap a gap, or escape to infinity — you’ve hit a discontinuity. Watch the pen try four different functions:
What you’re looking at: the pen traces left to right. If it ever has to jump, the function is not continuous there.
Only the first traces in one stroke. The others each force a lift — and each lift has a name you’ll learn below.
“Draw without lifting your pen” is the picture. Here it is as a single clean equation — the whole definition of continuity at a point lives in one line:
“Draw without lifting” is the feeling; here’s the precise check. For \(f\) to be continuous at a point \(a\), three things must all be true — and if any one fails, continuity fails. Run a function through the gate and watch each box get checked (or not):
The three checks are clearest when you operate them. Below, the curve heads toward height 2 at \(x=1\) — that’s the limit. Drag the point’s value up and down and watch the three boxes respond live. There’s exactly one spot where all three turn green:
What you’re looking at: the magenta curve heads to 2 from both sides. The teal dot is the actual value \(f(1)\) — you control its height. Continuity needs it to land exactly where the curve is heading.
When continuity fails, it fails in one of four ways. Each is a different “creature” with a recognizable shape and a specific broken condition. Meet them all:
The exam loves this: show a graph, ask what kind of discontinuity it has — and which part of the three-part test it violates. Diagnose each one:
Piecewise functions are continuous only if the two branches meet at the seam. Here the right branch is fixed at \(x^2\); the left is \(kx\). Drag \(k\) and slide the left branch until it meets the right one exactly at \(x=2\) — watch the gap shrink to zero:
What you’re looking at: the left branch \(kx\) (its slope changes with \(k\)) and the fixed right branch \(x^2\). The gap at the seam is the vertical distance between where they land at \(x=2\).
Left branch lands at \(2k\); right branch lands at \(4\). They’re equal exactly when \(2k=4\) — the value of \(k\) that makes \(f\) continuous.
One type of discontinuity is special: the removable one. The limit exists — the function just has the wrong value (or none) at a single point. You can patch it: redefine that one point to equal the limit, and the hole seals. These “find the value that makes \(f\) continuous” problems are an exam staple. Step through:
The recipe is always the same: compute the limit at the trouble point (using Lesson 2’s tools), then set the function’s value there equal to it. The hole fills.
Continuity is exactly limit equals value. It’s why substitution works for friendly functions — they’re continuous, so the limit IS \(f(a)\).
Value exists, limit exists, they match. Any one can fail on its own — that’s what makes the test diagnostic.
If the limit exists, the break is just one bad point — redefine it and the function becomes continuous. Jumps and asymptotes can’t be patched.
Polynomials are continuous everywhere; rational functions everywhere except where the denominator is zero. Knowing the type tells you where to look for trouble.
A function is continuous on an interval when it’s continuous at every point inside. At a closed endpoint, only the one side that exists needs to match — e.g. \(\sqrt{x}\) is continuous on \([0,\infty)\) because it’s right-continuous at \(0\).
Five problems — the three-part test, classification, and a solve-for-k.
Eight fast checks across the whole lesson.
The limit can exist while the function is still discontinuous — if \(f(a)\) is missing or has the wrong value. All THREE conditions are required, not just the limit.
Removable means the LIMIT EXISTS (both sides agree) — it’s just a hole. Jumps have disagreeing sides; infinite discontinuities blow up. Only removable ones can be patched.
A function can have a perfect limit and still be discontinuous because \(f(a)\) was defined as something else. Always check all three boxes.
For piecewise functions, continuity needs the left piece, right piece, AND the defined value to all agree at the seam. Set the one-sided limits equal — both of them.
“Discontinuous” isn’t automatically “removable.” Identify the type first — a jump or asymptote can never be patched by redefining one point.
A rational function is discontinuous wherever its denominator is zero — check those points specifically. Polynomials never have this problem; they’re continuous everywhere.