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Exploring Types of
Discontinuities

Discover when and why functions break: removable holes, jump gaps, and infinite asymptotes that shape the landscape of continuity.

AP Calculus BC

What Does Continuity Mean?

A function \( f \) is continuous at \( x = a \) when three conditions hold simultaneously: the function is defined, the limit exists, and they agree.

1
\( f(a) \) exists: the function has a real output at \( a \).
2
\( \displaystyle\lim_{x \to a} f(x) \) exists: left-hand and right-hand limits are equal.
3
\( \displaystyle\lim_{x \to a} f(x) = f(a) \): the limit equals the function value.

What Is a Discontinuity?

A discontinuity is any point where the function's graph cannot be drawn without lifting your pencil. There are three main types, each caused by a different failure of the continuity conditions.

Removable
A "hole" in the graph: the limit exists but \( f(a) \) is missing or mismatched.
Jump
The left-hand and right-hand limits both exist but are not equal.
Infinite
At least one side of the limit diverges to \( +\infty \) or \( -\infty \).

Three Discontinuity Types at a Glance

Removable hole at \( x=a \) Jump gap at \( x=a \) Infinite asymptote at \( x=a \)

Removable Discontinuity: The Hole

A function \( f \) has a removable discontinuity at \( x = a \) if the two-sided limit exists but does not equal the function value (or \( f(a) \) is undefined):

Why "removable"? You can redefine \( f(a) = L \) and the function becomes continuous. The classic example is a factor that cancels: \(\displaystyle f(x)=\frac{x^2-1}{x-1}\) has a hole at \( x=1 \) because the factor \( (x-1) \) cancels, leaving \( f(x)=x+1 \) everywhere except at \( x=1 \).

Jump Discontinuity: The Gap

A function \( f \) has a jump discontinuity at \( x = a \) if both one-sided limits exist but disagree:

Jump discontinuities are non-removable: no single value of \( f(a) \) can make both sides agree. Piecewise functions and step functions (like the greatest-integer function \( \lfloor x \rfloor \)) are common sources.

\( a \) \( L_1 \) \( L_2 \)

Infinite Discontinuity: The Asymptote

A function \( f \) has an infinite discontinuity at \( x = a \) when at least one one-sided limit diverges:

The graph shoots toward \( +\infty \) or \( -\infty \) near a vertical asymptote. Classic examples include \(\displaystyle f(x)=\frac{1}{x}\) at \( x=0 \) and \(\displaystyle f(x)=\tan(x)\) at \( x=\frac{\pi}{2} \). Like jump discontinuities, these are non-removable.

Removable vs. Non-Removable

The most important classification: can the discontinuity be "patched" by redefining a single point?

Removable
The two-sided limit \( L \) exists. Redefine \( f(a) = L \) to restore continuity. The graph has a hole, not a break.
Non-Removable
The two-sided limit either doesn't exist or is infinite. No single value of \( f(a) \) can make \( f \) continuous. Includes jump and infinite types.

Classification Flowchart

Is \( f \) continuous at \( x=a \)? No Does \( \lim_{x \to a} f(x) \) exist and is finite? Yes No Removable Do one-sided limits diverge to \( \pm\infty \)? No Yes Jump Infinite

How to Classify a Discontinuity

1
Find candidates. Identify where \( f \) might be discontinuous: where the denominator is zero, where a piecewise formula switches, or where the domain has a gap.
2
Compute one-sided limits. Evaluate \(\displaystyle\lim_{x\to a^-}f(x)\) and \(\displaystyle\lim_{x\to a^+}f(x)\) separately.
3
Check agreement. If both one-sided limits are equal and finite, the two-sided limit exists. If they differ, it's a jump. If either diverges, it's infinite.
4
Compare to \( f(a) \). If the limit exists but \( f(a) \neq L \) (or is undefined), it's removable. If \( f(a) = L \), the function is continuous.

Example 1: Removable Discontinuity

1
Factor: \(\displaystyle \frac{x^2-4}{x-2}=\frac{(x-2)(x+2)}{x-2}=x+2\) for \( x\neq 2 \).
2
Limit: \(\displaystyle\lim_{x\to 2}(x+2)=4\). Both one-sided limits equal 4.
3
Function value: \( f(2) \) is undefined (denominator zero, factor cancelled).
4
Conclusion: Limit exists but \( f(2) \) is undefined, so it's a removable discontinuity (hole at \( (2,4) \)).

Example 2: Jump Discontinuity

1
Left-hand limit: \(\displaystyle\lim_{x\to 3^-}(x+1)=\) 4.
2
Right-hand limit: \(\displaystyle\lim_{x\to 3^+}(2x-3)=\) 3.
3
Compare: 4 \(\neq\) 3, so the two-sided limit does not exist.
4
Conclusion: This is a jump discontinuity with a gap of size 1.

Example 3: Infinite Discontinuity

1 \( +\infty \) 0

Both \(\displaystyle\lim_{x\to 1^-}\frac{1}{(x-1)^2}=+\infty\) and \(\displaystyle\lim_{x\to 1^+}\frac{1}{(x-1)^2}=+\infty\). The function diverges on both sides, so this is an infinite (non-removable) discontinuity.

Example 4: Removable in Disguise

At first glance, \( g(0) \) is undefined (\( \frac{0}{0} \)). But the famous limit gives us:

Since the two-sided limit equals 1 (finite) but \( g(0) \) is undefined, this is a removable discontinuity. Redefine \( g(0)=1 \) to make \( g \) continuous everywhere.

Common Mistakes to Avoid

✕ Confusing Undefined with Infinite
\( f(a) \) being undefined does not mean an infinite discontinuity. If the limit is finite, it's removable even though \( f(a) \) doesn't exist.
✕ Forgetting to Check Both Sides
Computing only \(\displaystyle\lim_{x\to a} f(x)\) without splitting into left and right limits can hide a jump discontinuity.
✕ Cancelling Without Noting the Hole
After factoring \(\frac{(x-a)\cdot q(x)}{x-a}\), students forget that \( x=a \) is still excluded from the domain. The cancellation reveals the limit, not that \( f(a) \) exists.

Summary Comparison Table

Property Removable Jump Infinite
Two-sided limit Exists (finite) DNE (sides differ) DNE (diverges)
One-sided limits Both equal \( L \) Both finite, unequal At least one is \(\pm\infty\)
\( f(a) \) Undefined or \(\neq L\) May or may not exist Undefined
Graph feature Hole Gap / step Vertical asymptote
Can be fixed? Yes No No

Question 1 of 5

What type of discontinuity does \(\displaystyle f(x)=\frac{x^2-9}{x-3}\) have at \( x=3 \)?

Question 2 of 5

Consider \(\displaystyle h(x)=\begin{cases}2x & x < 1 \\ 5 & x \geq 1\end{cases}\). What type of discontinuity exists at \( x=1 \)?

Question 3 of 5

Which condition is sufficient to conclude a discontinuity at \( x = a \) is removable?

Question 4 of 5

What type of discontinuity does \(\displaystyle f(x)=\frac{1}{x-4}\) have at \( x=4 \)?

Question 5 of 5

Define \(\displaystyle g(x)=\begin{cases}\frac{x^2-x-6}{x-3} & x\neq 3 \\ 7 & x=3\end{cases}\). What type of discontinuity does \( g \) have at \( x=3 \)?