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 Common Core State Standards. It is aligned to Standard F-IF.7, which requires students to analyze, from a graph, where and how a function is discontinuous.
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 (\(x\to a^+\)).