Continuity, Part 2

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Anything You Want to Talk About?

Especially infinite limits, the first part of continuity, or other things from classes you couldn’t come to.

Saying lim_{x → a} f(x) = ∞ really means the limit is undefined? Yes, even though the notation looks like we’re saying that the limit has a value, “infinity” isn’t really a number, it’s more the idea of being big without any bound. So the “lim_{x → a} f(x) = ∞” notation is a shorthand way of saying that the limit is undefined, for the specific reason that f(x) gets unboundedly big near a.

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Continuity

From “Continuity over an Interval” and “The Intermediate Value Theorem” in section 2.4 of the textbook and this discussion.

Continuity and Intervals

Here is the same hypothetical function as you saw in the first continuity discussion:

Claim: this function is continuous on (-1,2) but not on [-1,2] nor [-1,2) nor (-1,2]. That is true, because the open interval (-1,2) doesn’t include either -1 or 2, and the function is in fact continuous at all other points between -1 and 2. But to be continuous on [-1,2] or [-1,2), the function would have to be continuous from the right at -1, i.e., the limit from the right would have to exist and equal f(-1). This one-sided limit does exist, but isn’t equal to f(-1). And to be continuous on (-1,2] the function and limit from the left would have to exist at x = 2, but neither does.

Key point: Continuity over an interval is mostly intuitive, except at the endpoints of the interval — pay attention to endpoints and whether the interval is open or closed.

Composite Function Theorem

Assume (as is the case) that sine and cosine are continuous functions. What are

and

In both cases, it’s tempting to to move the limit inside the cosine or sine functions. But we don’t actually have a limit law (like we did with the multiplication law, the constant multiple law, etc.) that lets us do that. But fortunately, we can use the composite function theorem to do it. For example...

For the sine example we can use algebra and the composite function theorem (that theorem can be used together with other things you know about limits):

In both cases it’s important to emphasize that we could only move the trig functions into the limits because the composite limit theorem applied. Any time you use a theorem, be sure its “fine print,” i.e., the requirements for it to apply, hold. In the case of the composite function theorem, the requirement is that the outer function is continuous at the inner function’s limit, and we did indeed check that that condition holds.

Intermediate Value Theorem

A picture of it (remember the tip for reading math about visualizing the ideas — this theorem lends itself well to that):

(Notice that this picture illustrates a particular case of the Intermediate Value Theorem, namely one in which it requires a function to cross the x axis. But visualizing a particular example of an idea is still valuable.)

What the Intermediate Value Theorem says makes excellent sense intuitively: If you know 2 values of a continuous function, the function has to hit all the values between those 2. But it shows up as an important step in a surprising number of other places.

For example...

Most outer space science fiction movies involve spaceships that fly faster than the speed of light.

According to Einstein, it takes infinite energy for an object with any mass at all to move at the speed of light.

Assume that science fiction spaceships have mass, and that even in science fiction movies it’s impossible to have an infinite amount of energy.

When a science fiction spaceship accelerates from standing still to above the speed of light, can its speed be a continuous function of time, or must it be discontinuous? Why?

The speed function must be either continuous or discontinuous, so consider each possibility in turn. If the speed is continuous, then the Intermediate Value Theorem says that sometime between when the spaceship was standing still and the time when it was moving faster than light, it must have been moving at the speed of light. But that’s impossible, so the only remaining option is for the speed to be discontinuous.

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Derivatives!

Read section 3.1 in the textbook.

Participate in this discussion.

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