Discontinuities (point, jump, infinite) and one-sided behavior
Discontinuities and One-Sided Behavior
Discontinuities and One-Sided Behavior is a topic in Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(L), P.2(M), which requires students to determine the types of discontinuities and describe the left- and right-sided behavior around them.
A discontinuity is a point where a graph breaks — removable (a hole), jump, or infinite (a vertical asymptote) — described by the one-sided behavior on each side.
Theory
A function is discontinuous at a point where its graph breaks. There are three types:
- Removable (point): a single missing point — a hole. It appears when a factor cancels, as in \(\dfrac{x^2-4}{x-2}\) at \(x=2\).
- Jump: the left and right pieces sit at different heights, so the graph “jumps.” Common in piecewise functions.
- Infinite: the function grows without bound near the point — a vertical asymptote, as in \(\dfrac{1}{x-3}\) at \(x=3\).
To pin down the behavior near a break, examine the one-sided behavior: what \(f(x)\) does as \(x\) approaches from the left (\(x\to a^-\)) and from the right (\(x\to a^+\)).
One-sided behavior is written with superscript signs:
A discontinuity at \(x=a\) in a rational function comes from a zero of the denominator; whether it is a hole or an asymptote depends on whether the factor cancels.
How to classify a discontinuity
- Find where the function breaks (zero denominator, or a boundary in a piecewise rule).
- Factor a rational function: a canceling factor \(\Rightarrow\) hole; a non-canceling one \(\Rightarrow\) vertical asymptote.
- Check one-sided values: equal but with a hole \(\Rightarrow\) removable; finite and different \(\Rightarrow\) jump; unbounded \(\Rightarrow\) infinite.
Factor and cancel; the factor \((x-2)\) divides out.
| \(f(x)\) | \(=\) | \(\dfrac{(x-2)(x+2)}{x-2}\) |
| \(=\) | \(x+2,\quad x\neq 2\) |
The graph is the line \(y=x+2\) with a single missing point at \((2,4)\) — a removable (point) discontinuity, a hole.
The denominator is zero at \(x=3\) but the numerator is not, so the factor does not cancel.
| \(x\to 3^-\) | \(\Rightarrow\) | \(f(x)\to-\infty\) |
| \(x\to 3^+\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
This is an infinite discontinuity — a vertical asymptote at \(x=3\).
Compare the one-sided values at \(x=1\).
| \(\text{left: } x-1\to\) | \(1-1=0\) | |
| \(\text{right: } x+2\to\) | \(1+2=3\) |
The left value \(0\) and right value \(3\) differ, so there is a jump discontinuity of size 3 at \(x=1\).
The denominator \((x-4)^2\) is positive on both sides and shrinks to 0, so the quotient grows large and positive from each side.
| \(x\to 4^-\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
| \(x\to 4^+\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
Both sides go to \(+\infty\): an infinite discontinuity where the curve rises on both sides of \(x=4\).
Common pitfalls
Frequently asked questions
What are the three types of discontinuity?
Removable (a hole), jump (left and right pieces at different heights), and infinite (a vertical asymptote).
What is a removable discontinuity?
A single missing point — a hole — that appears when a factor cancels, such as \(\dfrac{x^2-4}{x-2}\) at \(x=2\).
How do you tell a hole from a vertical asymptote?
Factor the rational function. If the zero factor in the denominator cancels with the numerator you get a hole; if not, you get a vertical asymptote.
What does one-sided behavior mean?
How the function behaves as \(x\) approaches a point from just one side: from the left (\(x\to a^-\)) or from the right (\(x\to a^+\)).