Topic 1.10 key terms
- continuous at a point
- f is continuous at x = c when f(c) exists, the limit of f(x) as x approaches c exists, and that limit equals f(c).
- one-sided limit
- The height f(x) heads toward as x approaches c from one side only: from the left (x → c⁻) or from the right (x → c⁺).
- jump discontinuity
- Both one-sided limits exist but are different, so the limit does not exist. No single value at c can make f continuous there.
- removable discontinuity
- The limit at c exists, but f(c) is missing or different from it. Defining or redefining f(c) as the limit removes the discontinuity.
- infinite discontinuity
- A discontinuity due to a vertical asymptote: at least one one-sided limit is infinite, so no value of f(c) can make f continuous.
- removing a discontinuity
- If the limit exists at a discontinuity, define or redefine the function's value there to equal the limit.
- piecewise boundary condition
- At a boundary, a piecewise function is continuous when the expressions on both sides give the same value there and that value is the function's value at the boundary.
Transcript
920 words · 3 parts
The line in the Pacific
In February 2007, a formation of F-22 Raptors, America’s newest stealth fighters, took off from Hawaii and headed west across the Pacific toward Japan, on their very first trip overseas. A few hours out, over nothing but open ocean, they crossed the International Date Line. Right at that line, their computers crashed. Navigation went down, and so did some of their radios. Rebooting didn’t help, so the pilots stuck close to their tankers and followed them all the way back to Hawaii.
What I love about this story is that the jets themselves never jumped, not even by a centimeter. But one number describing them did jump. The computer stores west longitudes as negative numbers and east ones as positive, so its number goes from negative one eighty to positive one eighty in a single instant. Those two numbers name the very same line on the globe.
The reports don’t say which line of code failed, but a leap like that is exactly the kind of surprise software has to be ready for. By the end of this lesson, you’ll be able to look at a function and say exactly where it breaks, what kind of break it is, and whether you can fix it.
At the very end of the lesson, you’re going to fix this one yourself. To fix a break, we first need to know exactly what no break means, and the jets’ longitude is the perfect place to start.
Two sides that don’t agree
Let’s graph that number. I’ll start the clock an hour before the jets reach the line, so they cross it at t equals one. Up the side is the longitude the computer stores, in degrees. As the jets close in on the line, the longitude slides down toward negative one eighty, and the closer they get, the closer it gets. In limit language, the limit of lambda, our longitude, as t approaches one from the left is negative one eighty.
Now come at the line from the other side, from just after the crossing, and the longitude heads up toward positive one eighty instead. That’s the limit from the right, positive one eighty, a full three hundred sixty degrees away from the left side. Right at the line itself, the computer has to store one number, and it stores positive one eighty, which is this solid dot.
I want you to make a call before we go on. Is there one value we could store right at the line that would join these two pieces into one unbroken graph? Don’t take my word for it. Drag the dot to any height you like, and see whether the two pieces ever join.
When the two sides head to two different heights, we call it a jump discontinuity, and you can see where the name comes from. The two one-sided limits disagree, so the limit as t approaches one doesn’t exist at all. A limit has to be one single height that both sides are heading toward.
That’s why no value at the line can save it. One dot can fill a missing point, but it can’t drag two pieces together. Hold on to that, because we’re coming back to these jets at the end with a better idea.
The brightest point
This next break is a completely different kind, and it’s hiding in the brightest spot of this picture. This is a green laser shining through a slit thinner than a hair, and on the wall you get this famous pattern, a bright stripe in the middle and fainter stripes fading away on both sides.
Physicists have a formula for how bright the wall is at each point, compared with the center. The brightness B is sine x over x, all squared, where x is a scaled distance from the center. Now ask the formula about the very center, where x is zero, and you get sine of zero over zero, which is zero over zero, and that isn’t a number at all.
Look at what that does. The real light on the wall is perfectly bright right there, but a computer that just runs the formula gets “not a number” and paints a black line straight through the brightest stripe. The formula has a hole, and it’s sitting exactly at its highest point.
Let’s find out what the brightness is heading toward. This detector on a rail reads the brightness, so slide it in toward the center, once from the left and once from the right, and keep your eye on the readout.
Both sides head to one, so the limit as x approaches zero is one. You’ve actually seen why. Back when we squeezed sine x over x, we showed it heads to one, so its square heads to one too.
The value at zero is missing, but the limit is sitting right there at one, and we call a break like that a removable discontinuity, because you can repair it. A programmer in a hurry might patch it by telling the computer that whenever x is zero, the brightness is zero. Now there is a value at zero, but the stripe still has a dark seam, because zero isn’t what the nearby brightness is heading toward.
Now fix it properly. The programmer’s line says if x is zero, return zero, so change that number to the one value that makes the seam disappear completely. That’s what it means to remove a discontinuity. You define the value at the break to be the limit, and the break is gone.