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Continuity is the formal version of "you can draw it without lifting your pen." Watch the pen do it, then watch it fail.
Formally, three things have to be true at a point: the function is defined there, the limit exists there, and the two are equal. Miss any one and the pen lifts.
The useful part: each of the last three episodes broke exactly one condition. The hole in episode 2 — the limit exists, but f(1) doesn't. The jump in episode 3 — f(1) exists, but the limit doesn't. And a point floating off its own curve — both exist, and they disagree.
One warning: continuous does not mean smooth. The absolute value function has a perfect corner at zero and never lifts the pen. That corner is exactly where the derivative breaks in episode 7.
Calculus #6 of 113. Full playlist on my channel
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2026/09/13
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