One Point Is Not an Interval

You'll learnto read continuity on an interval as a claim about every point of it, to turn that infinite check into a short finite one by naming the function family and listing its suspect points, to check a closed endpoint from inside the interval only, and to recognise the sample that steps straight over the failure.

Continuity on an interval is continuity at every point of it, which is a far stronger claim than continuity at one. Two things follow, and they are the whole topic. Because you cannot test infinitely many points, the argument runs backwards: name the family the function belongs to, then hunt the few places that family can fail. And because a closed interval's endpoint has only one side inside the interval, the check there is one-sided by design — demanding a limit from both sides would be asking about inputs the claim never mentioned.

8 STEPS · 6 QUICK CHECKS · EVERY POINT, AND THE ENDS ONE-SIDEDLY · v1

continuous on I means continuous at EVERY point of I · and a closed end is checked from inside

Before you start

What you're looking at

One graph, with a shaded band across the inputs from 1 to 5. That band is the interval the claim is about; everything outside it is not part of the question.

The question

Is the function continuous at every single input inside the band — not at one convenient input, and not merely at the two ends?

Watch for

A hollow ring and a solid disc landing together mark one input passing the test, exactly as in Topic 1.11. A green stretch of curve marks inputs already certified. Where the green stops is where the interval claim breaks.

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