Why Classify Discontinuities?
A discontinuity tells us that a function fails to be continuous at a point, but it does not tell us how it fails. Near a point, a graph might approach one finite height from both sides while the function is missing that value. It might approach different finite heights from the left and right. Or its values might grow without bound. These behaviors are different, and one-sided limits allow us to tell them apart.
For an interior point \(a\), the classification depends first on the left-hand and right-hand limits, and then on the actual function value \(f(a)\). A plotted point alone cannot show the full behavior nearby, and a limit alone does not say whether the function is defined at \(a\).
Continuity and One-Sided Limits
Suppose \(f\) is defined in a neighborhood of \(a\), except possibly at \(a\). The function is continuous at \(a\) if \(f(a)\) is defined, the two-sided limit \(\lim_{x\to a}f(x)\) exists, and that limit equals \(f(a)\). For an interior point, the two-sided limit exists with finite value \(L\) exactly when both one-sided limits exist and equal \(L\).
The categories can be stated precisely:
- Removable discontinuity: The finite two-sided limit \(\lim_{x\to a}f(x)=L\) exists, but \(f(a)\) is undefined or \(f(a)\ne L\). The function approaches one finite value from both sides, but its value at the point does not fit that behavior.
- Jump discontinuity: Both one-sided limits exist and are finite, but they are unequal: \(\lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x)\). The graph approaches different finite heights from the two sides.
- Infinite discontinuity: At least one one-sided limit is \(+\infty\) or \(-\infty\). The function grows without bound in magnitude from that side as \(x\) approaches \(a\).
These categories describe common forms of discontinuity, but they do not cover every possible failure of a limit. For example, if the function oscillates near \(a\) without approaching a finite value or either infinity, it does not fit any of these three definitions. Do not force a classification when the one-sided limits do not meet its conditions.
A Short Justification: Why the Two Sides Matter
The two-sided limit at an interior point describes behavior from both directions. The following argument explains why checking both one-sided limits gives exactly the test needed for a finite two-sided limit.
Suppose \(\lim_{x\to a^-}f(x)=L\) and \(\lim_{x\to a^+}f(x)=L\). For any \(\varepsilon>0\), the definition of each one-sided limit gives a positive distance on its respective side within which \(|f(x)-L|<\varepsilon\).
Let \(\delta\) be the smaller of those two positive distances. If \(0<|x-a|<\delta\), then \(x\) is either to the left or to the right of \(a\), and the corresponding one-sided limit gives \(|f(x)-L|<\varepsilon\). Thus \(\lim_{x\to a}f(x)=L\).
If \(\lim_{x\to a}f(x)=L\), the defining inequality holds for all \(x\) sufficiently close to \(a\), including those with \(x<a\) and those with \(x>a\). Therefore both one-sided limits equal \(L\).
So if the finite one-sided limits are unequal, no finite two-sided limit can exist. If they agree, there is a finite two-sided limit, and the remaining continuity question is whether \(f(a)\) is defined and equal to it.
Worked Example: A Removable Discontinuity
Worked Example: A Removable Discontinuity
Classify the discontinuity of \(f(x)=\frac{x^2-1}{x-1}\) for \(x\ne1\), with \(f(1)=5\).
First find the limit for \(x\ne1\). Factor the numerator:
The cancellation is valid for \(x\ne1\), which is sufficient when taking a limit at \(1\). Thus the one-sided limits are
The two-sided limit is therefore \(2\), while the assigned function value is \(f(1)=5\). Since the finite limit exists but does not equal the function value, \(f\) has a removable discontinuity at \(x=1\).
Check: For every \(x\ne1\), the simplified expression gives \(f(x)=x+1\), whose values approach \(2\) as \(x\) approaches \(1\). The separate value \(f(1)=5\) does not change either one-sided limit, confirming the classification.
Worked Example: A Finite Jump
Worked Example: A Finite Jump
Classify the discontinuity at \(x=0\) for
For \(x<0\), the relevant rule is \(x+1\). Hence
For \(x>0\), the rule is \(x^2+2\), so
Both one-sided limits are finite, but \(1\ne2\). Therefore the two-sided limit does not exist and the function has a jump discontinuity at \(x=0\). The function value is \(f(0)=0^2+2=2\), but the value at the point cannot make the two unequal one-sided limits agree.
Check: Approaching from the left gives height \(1\), while approaching from the right gives height \(2\). These are exactly the two distinct finite values found by the one-sided calculations.
Worked Example: An Infinite Discontinuity
Worked Example: An Infinite Discontinuity
Classify the discontinuity of \(g(x)=\frac{x+2}{x-3}\) at \(x=3\).
Near \(x=3\), the numerator \(x+2\) approaches \(5\), which is positive. The denominator \(x-3\) approaches zero, and its sign depends on the side:
- If \(x<3\), then \(x-3<0\), so a positive numerator divided by a negative denominator becomes arbitrarily large in the negative direction.
- If \(x>3\), then \(x-3>0\), so a positive numerator divided by a positive denominator becomes arbitrarily large in the positive direction.
Therefore
At least one one-sided limit is infinite—in fact, both are—so \(g\) has an infinite discontinuity at \(x=3\).
Check: At \(x=2.9\), \(g(2.9)=4.9/(-0.1)=-49\); at \(x=3.1\), \(g(3.1)=5.1/0.1=51\). These sample values are consistent with the one-sided signs. The classification follows from the limits, not just from these examples.
A Reliable Classification Process
For an interior point \(a\), organize the work in the same order each time. This prevents the function value at the point from distracting from what happens nearby.
Use the formula that applies when \(x<a\), or analyze the behavior on the left side.
Use the formula that applies when \(x>a\), or analyze the behavior on the right side.
Unequal finite limits indicate a jump. A one-sided limit of \(+\infty\) or \(-\infty\) indicates infinite behavior. Equal finite limits give a finite two-sided limit.
If the one-sided limits agree at a finite value \(L\), check whether \(f(a)\) is defined and whether \(f(a)=L\). If either condition fails, the discontinuity is removable; if both hold, the function is continuous at \(a\).
Common Pitfall: Using the Point Value Instead of the Limits
A frequent error is to decide that a function is continuous because its assigned value agrees with one of the one-sided limits. In the jump example, \(f(0)=2\), and the right-hand limit is also \(2\). That agreement is not enough: the left-hand limit is \(1\), so the two-sided limit does not exist. The correct test requires both sides to agree with each other before comparing their common value with \(f(0)\).
There is a related mistake when simplifying a quotient. In the removable example, canceling \(x-1\) gives \(x+1\) for \(x\ne1\). It would be wrong to conclude from this cancellation that the original function’s value at \(1\) is \(2\): the original definition explicitly gives \(f(1)=5\). Cancellation identifies the nearby behavior and therefore the limit; it does not change a separately assigned point value.
Why the Classification Is Useful
Classification turns a general statement—“the function is discontinuous”—into a description of its local behavior. A removable discontinuity has a common finite limiting height. A jump has two distinct finite one-sided heights. An infinite discontinuity involves unbounded behavior from at least one side. This distinction helps you interpret formulas and graphs accurately and choose the right limit test for a given point.
Keep the categories tied to their definitions. Unequal finite one-sided limits are a jump, not a removable discontinuity. A finite common limit with a mismatched point value is removable, not a jump. A one-sided infinite limit is infinite behavior, even if the other side approaches a finite value. If neither one-sided limit exists in one of these ways, report that the behavior does not fit these three classifications rather than guessing.
Check Your Understanding
For each question, identify the relevant one-sided limits before deciding how the point should be classified.
- A function has left- and right-hand limits equal to \(4\), but \(f(a)=7\). What type of discontinuity occurs?
- The left-hand limit is \(-2\) and the right-hand limit is \(3\). What type of discontinuity occurs, and does the value \(f(a)\) affect that classification?
- The left-hand limit is \(-\infty\) and the right-hand limit is \(6\). How should the discontinuity be classified?
- Both one-sided limits equal \(5\), and \(f(a)=5\). Is there a discontinuity at \(a\)?
- Why does canceling a common factor in a formula for \(x\ne a\) not, by itself, determine the original function’s value at \(a\)?
