Topic 1.12 key terms
- continuous on an interval
- A function is continuous on an interval if it is continuous at every point of the interval: no holes, jumps or vertical asymptotes anywhere in it. A corner is fine.
- continuity on a closed interval [a, b]
- Continuous at every point of (a, b), plus one-sided at the ends: the limit from the right at a equals f(a), and the limit from the left at b equals f(b).
- continuous on its domain
- Polynomial, rational, power, exponential, logarithmic and trigonometric functions are continuous at every point of their domains; they can only break at inputs they can't take.
- Intermediate Value Theorem (IVT)
- If f is continuous on [a, b] and d is a number between f(a) and f(b), then there is at least one number c between a and b with f(c) = d.
- existence, not location
- The IVT guarantees that some c exists. It doesn't say where c is, and it doesn't say there is only one.
- IVT justification
- Full credit: why f is continuous on the closed interval; the target between the two endpoint outputs, with their values; the conclusion that at least one c in the open interval has f(c) = the target.
- when the IVT is silent
- If f isn't continuous on the interval, or the target isn't between the endpoint outputs, the IVT guarantees nothing, which is not the same as saying the value never happens.
Transcript
231 words · 1 parts
The wobbly table
You’re on a terrace on the Amalfi coast, about an hour before sunset, with the sea way down below you and a cold lemon soda on a little marble table. You reach for your drink, you lean on the table just a little, and the whole thing does this. Get down at floor level and you can see the problem, because one foot is hanging about six millimeters above the stone, so the table rocks every time you touch it.
Most people fold up a napkin and jam it under the short leg, but there’s a much better trick, and you don’t need a napkin at all. You just turn the table, right where it stands, nice and slow, and watch the gap under that foot get thinner and thinner, until all four feet are on the stone.
That isn’t luck. On a floor with no steps in it, as long as it isn’t too steep, a table like this always has a spot within a quarter turn where all four feet touch the ground.
By the end of this lesson, you’ll be able to prove that before you even try it, and you’ll know exactly what kind of floor could break the trick. The proof is one of the big theorems of calculus, and it starts with a simple question about this table. Which feet is it actually rocking on?