A function is continuous at a point when its graph has no break, jump, hole, or vertical blow-up there.
To classify a discontinuity, compare the left-hand limit, right-hand limit, and the actual value of the function.
1. Continuity at a point
A function \(f\) is continuous at \(x=a\) if all three conditions hold:
\[
f(a)\text{ is defined}
\]
\[
\lim_{x\to a}f(x)\text{ exists}
\]
\[
\lim_{x\to a}f(x)=f(a)
\]
If one of these conditions fails, the function is discontinuous at \(x=a\).
2. Removable discontinuity
A removable discontinuity happens when the limit exists, but the point value is missing or wrong.
\[
\lim_{x\to a}f(x)=L
\]
but either \(f(a)\) is undefined or \(f(a)\ne L\).
Example:
\[
f(x)=\frac{x^2-1}{x-1}
\]
At \(x=1\), direct substitution gives \(\frac00\). Factor:
\[
\frac{x^2-1}{x-1}
=
\frac{(x-1)(x+1)}{x-1}
=
x+1,
\qquad x\ne1
\]
Therefore:
\[
\lim_{x\to1}\frac{x^2-1}{x-1}=2
\]
The original function is undefined at \(x=1\), so there is a removable hole.
3. Jump discontinuity
A jump discontinuity occurs when the left and right limits are finite but different:
\[
\lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x)
\]
In this case, the two-sided limit does not exist.
For example, a piecewise function may approach one height from the left and another height from the right.
4. Infinite discontinuity
An infinite discontinuity happens when the function grows without bound near the point:
\[
f(x)\to+\infty
\quad\text{or}\quad
f(x)\to-\infty
\]
Example:
\[
f(x)=\frac{1}{(x-2)^2}
\]
As \(x\to2\), the denominator approaches \(0\) while staying positive. The function grows upward without bound:
\[
\lim_{x\to2}\frac{1}{(x-2)^2}=+\infty
\]
The graph has a vertical-asymptote type of discontinuity.
5. Oscillatory or inconclusive behavior
Some functions do not settle to one value near the point. A common example is:
\[
f(x)=\sin\left(\frac1x\right)
\]
As \(x\to0\), the function keeps oscillating between \(-1\) and \(1\).
It does not approach a single value.
6. Summary table
| Type |
Left and right limits |
Point value |
Graph sign |
| Continuous |
Equal |
Equal to the limit |
No break |
| Removable |
Equal |
Missing or different |
Open circle or wrong filled point |
| Jump |
Different finite values |
May or may not be defined |
Sudden jump in graph height |
| Infinite |
One or both sides unbounded |
Usually undefined |
Vertical-asymptote behavior |
| Oscillatory |
No stable value |
May or may not be defined |
Rapid repeated oscillation |
7. Piecewise functions
For a piecewise function, check the rule from the left, the point value, and the rule from the right separately.
\[
f(x)=
\begin{cases}
g(x),&x
a
\end{cases}
\]
Then compare:
\[
\lim_{x\to a^-}g(x),
\qquad
c,
\qquad
\lim_{x\to a^+}h(x)
\]
If the two one-sided limits agree but \(c\) is missing or different, the discontinuity is removable.
If the two one-sided limits disagree, it is a jump.
8. Common mistakes
- Do not confuse \(f(a)\) with \(\lim_{x\to a}f(x)\).
- A function can have a limit at a point where it is not defined.
- A hole is removable only when the left and right limits agree.
- A jump occurs when the left and right limits are different finite values.
- An infinite discontinuity means the graph grows without bound near the point.
- Always check both sides before deciding the type of discontinuity.
