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Function

Would you describe this as a "hole" at x = 2? The reason why I ask is that I have learned that holes are where the function is not defined. But the function is defined at x = 2, or f(2)=1. Can a hole still exist if the function is defined at that x value?

Jisang222
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1 Answers1

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What CollegeBoard thinks a hole is

It is common for definitions to vary in mathematics, so the first step to answering what a word means, in case of doubt, is always to ask what the person using it defined it to mean. Unfortunately, in this case, that means we have to consult the AP Precalculus Course and Exam Description.

Here, holes are only discussed in the context of rational functions. It says that they occur when a factor $(x-c)$ occurs more times in the denominator than the numerator, and that the hole is located at the point $(c,L)$ when $\lim_{x \to c} f(x) = L$, but it's unclear whether these are definitions or observations.

The likeliest interpretation from there is that CollegeBoard promises not to discuss holes in the case that the function is not a rational function. (For a rational function, the diagram in the question can never occur; a rational function will always be undefined at such a point.)

If such a graph does appear on the AP Precalculus exam, I would lean toward calling it a hole for two reasons:

  • It's also a wacky situation where the limit exists, and the limit is the only thing that the course description even mentions that applies here.
  • It fits better with what the rest of the world thinks, see below.

What mathematicians think a hole is

The problem is that serious mathematicians would not call something a hole. They would use words like "removable discontinuity". The word "hole" would be an informal description of the actual mathematical term.

Discontinuities are classified as "removable", "step", and "essential", and the picture in the diagram is definitely without question a removable discontinuity: it's a discontinuity where the limit exists, it's just that the function value is not equal to the limit.

There is disagreement about what the word "removable discontinuity" is, but the disagreement tends to go the other way:

  • Some people are happy to say that it's a removable discontinuity even if the function is not defined at all at the point.
  • Some people say that a function can only have a discontinuity at a point if that point is in the domain to begin with. With this interpretation, asking "what kind of discontinuity does $f(x)$ have at $x=2$?" is just as silly as asking "what kind of discontinuity does $f(x)$ have at $x=\text{purple}$?"

So it would be reasonable to say that only the situation in the question is a hole, or to say that it's a hole in that case and also if the function is undefined at that point. By using the term for rational functions, CollegeBoard has probably decided that they're in the second camp.

Or they just haven't thought about it, in which case it could go either way.

Misha Lavrov
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