Concept / Calculus
Not every break fails in the same way.
Continuity at a point requires a defined value, an existing limit, and agreement between them. When one condition fails, the pattern of failure determines the type of discontinuity.
The continuity contract
A function f is continuous at a when all three statements hold:
- f(a) is defined.
- limx→a f(x) exists.
- limx→a f(x) = f(a).
The first condition concerns the point. The second concerns nearby behavior from both sides. The third connects them. Testing these separately prevents a hole, jump, asymptote, or oscillation from being treated as the same phenomenon.
Four failure signatures
| Type | One-sided behavior | Can a single point repair it? |
|---|---|---|
| Removable | Finite left and right limits agree | Yes, define the point to equal the limit |
| Jump | Finite left and right limits disagree | No |
| Infinite | Magnitude grows without finite bound | No |
| Oscillatory | Values do not settle to one result | No |
Representative models
Removable: (x²-1)/(x-1) = x+1 for x ≠ 1
Jump: f(x) = -1 for x<0 and 1 for x≥0
Infinite: f(x) = 1/x² near x=0
Oscillatory: f(x) = sin(1/x) near x=0
A graph is useful for recognizing these signatures, but the classification comes from limits. In particular, finite sampling cannot prove that oscillation has stopped or that growth remains bounded closer to the point.
One changed value cannot repair neighborhood behavior.Redefining f(a) can fill a removable hole because the nearby values already agree. It cannot reconcile unequal one-sided limits, cap an infinite asymptote, or force an oscillating neighborhood to settle.