Continuity Is the Ring and the Disc in the Same Place

You'll learnto state all three continuity conditions, to watch each of them fail while the other two hold, to solve for a constant that forces a piecewise function to be continuous, and to recognise the two-out-of-three check that is the standard way to get this topic wrong.

Topic 1.2 spent eight steps separating two objects: the ring, which marks where the outputs are heading, and the disc, which marks what the function actually is at the point. Continuity is the name for the case where those two land in the same place. Written out it is three conditions — the value exists, the limit exists, and the two agree — and each of them can fail while the other two hold. The third is the one that gets skipped, because the first two are the ones you naturally compute.

8 STEPS · 6 QUICK CHECKS · VALUE, LIMIT, AND THEY MUST AGREE · v1

a value at the point · a limit at the point · and the two of them equal

Before you start

What you're looking at

One graph with a marked input. Two objects sit above that input: a hollow ring at the height the outputs approach, and a solid disc at the height the function actually takes there.

The question

Are both objects present, and are they in the same place? Those two questions are the whole definition of continuity at a point.

Watch for

Each step removes exactly one of the three requirements. A ring with no disc, two rings with no agreement, and a ring and a disc at different heights are three different failures, and only the third one lets every calculation come out fine.

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