1.10 Exploring Types of Discontinuities
Keywords
Removable Discontinuity 可去间断点 (Hole) • Jump Discontinuity 跳跃间断点
Infinite Discontinuity 无穷间断点 (Vertical Asymptote) • Continuous 连续
What is a Discontinuity?
A function is discontinuous at a point \(x=c\) if there is any "break," "hole," or "gap" in the graph at that point. Formally, if a function is not continuous, it must fall into one of the three categories below.
核心要点：Three Types of Discontinuity
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Removable Discontinuity (Hole): The limit \(\lim\limits_{x \to c} f(x)\) exists, but it does not equal the function value \(f(c)\) (either \(f(c)\) is undefined or it is defined at a different height).
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Jump Discontinuity: The one-sided limits \(\lim\limits_{x \to c^-} f(x)\) and \(\lim\limits_{x \to c^+} f(x)\) both exist as finite numbers, but they are not equal.
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Infinite Discontinuity: The function values grow without bound as \(x\) approaches \(c\) from one or both sides (creating a Vertical Asymptote).
1. Removable Discontinuity (Holes)
Algebraically, these occur when a factor in the denominator cancels out with a factor in the numerator (the \(0/0\) case we saw in Section 1.6).
In the graph above, \(\lim\limits_{x \to 2} f(x) = 4\), but \(f(2) = 1\). Because the limit exists but doesn't match the point, it is removable.
2. Jump Discontinuity
These are most common in piecewise functions. The graph literally "jumps" from one \(y\)-value to another at \(x=c\).
Here, \(\lim\limits_{x \to 2^-} f(x) = 3\) and \(\lim\limits_{x \to 2^+} f(x) = 1\). Since \(3 \neq 1\), the general limit does not exist.
3. Infinite Discontinuity (Vertical Asymptotes)
Algebraically, these occur when the denominator approaches \(0\) but the numerator approaches a non-zero constant (the \(k/0\) case).
At \(x=2\), the function shoots off to \(+\infty\) from the right and \(-\infty\) from the left.
Extension: Why "Removable"?
The term "removable" is used because you could make the function continuous just by redefining a single point. If you "plug the hole" by setting \(f(c) = L\), the discontinuity vanishes. In contrast, you cannot make a Jump or Infinite discontinuity continuous by changing just one point—you would have to shift the entire branch of the graph!