1. What a Discontinuity Means
A discontinuity occurs when a function's graph breaks, jumps, or behaves unexpectedly at a point. Even if the function works fine elsewhere, a single troublesome point can make it discontinuous there.
In simple terms: something goes wrong when approaching that point — the value doesn't match, or the behaviour is unpredictable.
2. Why Discontinuities Occur
Discontinuities happen when the function value is missing, the left- and right-hand limits disagree, or the output blows up. Understanding these cases helps identify what type of break the graph has.
3. Removable Discontinuity
This happens when the limit exists but the function value is missing or different from the limit. The graph has a hole.
3.1. Example
f(x) = \frac{x^2 - 4}{x - 2}
Limit at x = 2 is 4, but f(2) is not defined. So there's a hole at x = 2.
4. Jump Discontinuity
A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal. The graph suddenly jumps from one value to another.
4.1. Example
f(x) = \begin{cases}1, & x < 0 \\ 3, & x > 0 \end{cases}
Left-hand limit is 1 and right-hand limit is 3 → jump at x = 0.
5. Infinite Discontinuity
This occurs when the function approaches ±∞ near the point. The graph shoots upward or downward without bound — often at vertical asymptotes.
5.1. Example
f(x) = \frac{1}{x}
As x → 0, the function becomes unbounded, giving an infinite discontinuity at x = 0.
6. Oscillating Discontinuity
Here, the function swings back and forth rapidly near a point, preventing the limit from existing. The graph becomes extremely wavy as x approaches the point.
6.1. Example
f(x) = \sin \frac{1}{x}
Near x = 0, the values oscillate endlessly between -1 and 1, so the limit does not exist.
7. Infinite Jump-Like Discontinuity (Special Case)
Sometimes the left-hand and right-hand limits go to ±∞ but with different signs. This causes a sharper, extreme jump between vertical asymptotes.
8. Graphical Summary
Each discontinuity type has its own visual pattern:
- Removable: hole in the curve
- Jump: sudden vertical jump
- Infinite: vertical blow-up
- Oscillating: rapid swings
Recognizing these patterns helps identify the type of discontinuity quickly.