Continuity

Stage 18 of 23 Strand 2 of 2 3 lessons

3 illustrated lessons, each teaching the why before the how.

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Continuity at a Point #

Continuous where the limit meets the value.

A function is continuous where its limit and its value agree

You can draw this without lifting the pencil. That is continuity, informally.

Formally: the value the function approaches must equal the value it actually takes.

All three must hold. Any one of them failing breaks the curve at that point.

Now you

The limit is 4 and f(2) is 4. Is f continuous at 2?

The limit is 4 but f(2) is 7. Is f continuous at 2?

A curve breaks by a hole, a jump or a blow-up.

A curve can break at a hole, at a jump, or where it runs off to infinity

A removable discontinuity is one missing point: fill the hole and the curve is mended.

A jump discontinuity has two different one-sided limits, and no single point can mend it.

At an infinite discontinuity the curve runs off to infinity along a vertical asymptote, so no limit exists there.

Now you

The left limit is 1 and the right limit is 4. Which kind of discontinuity?

The curve runs off to infinity along a vertical asymptote. Which kind of discontinuity?

No skipping a height on a continuous curve.

The Intermediate Value Theorem: a continuous curve takes every height in between

At x = 0 this curve is below the axis. At x = 2 it is above the axis.

To get from below to above without lifting the pencil, it must cross zero.

So a sign change between two values guarantees a root between them.

Now you

f(1) = 2 and f(2) = 6, and f is continuous. Must there be a root between?

f(0) = −1 and f(4) = 1, and f is continuous. Is there a root between?