Continuity Preview

Calculus Readiness · Axiom Academy

A function is continuous when you can draw it without lifting your pen — and limits make that idea exact. 1. Continuous: One Unbroken Stroke Informally, a function is continuous if you can draw its graph without lifting your pen. Watch the pen below trace the whole curve in a single motion — no holes to skip over, no gaps to jump, no place it has to leave the page. That single stroke is continuity. That picture is great intuition, but math needs something exact. The precise version uses limits — which is why continuity is the perfect place to land a unit on limits. (1) the value exists · (2) the limit exists · (3) they agree At the test point x = a , all three conditions pass at once: 2. A Hole: Removable Discontinuity Now the first break. The pen below glides along the line, but at x = a there is no point to land on — an open circle, a hole . The limit is perfectly well-behaved (the curve clearly heads for one height), yet the function has no value there. Condition 1 fails. It is called removable because filling that single point would repair everything — define f(a) to equal the limit and the curve becomes continuous. We can patch the hole. Worked check — apply the three conditions Conclusion: f is not continuous at x = 2 — a removable discontinuity. Redefining f(2) = 4 would patch it. (The classic behaves the same way: undefined at 2 , but heading for 4 .)

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