Discontinuous Function - Types, Graph & Examples
Discontinuous Function - Types, Graph & Examples
TL;DR
A discontinuous function is one whose graph cannot be drawn without lifting your pen — it has at least one break, hole, jump, or vertical asymptote. This article walks through the three main types — removable, jump, and infinite discontinuities — shows how to read each off a graph, and works through six examples with the mistakes to avoid.
What Is A Discontinuous Function?
A discontinuous function is a function that is not continuous at one or more points in its domain — meaning its graph has at least one break, hole, jump, or vertical asymptote. Where a continuous function can be traced in one unbroken stroke, a discontinuous function forces the pen off the paper at least once.
Formally, a function f is continuous at x=a when three conditions all hold: f(a) exists, the limit ( \lim_{x \to a} f(x) ) exists, and those two values are equal. If any one of the three fails, f is discontinuous at x=a.
| Symbol | Meaning |
|---|---|
| f(a) | the function's value at the point a |
| ( \lim_{x \to a^-} f(x) ) | the left-hand limit (approaching a from below) |
| ( \lim_{x \to a^+} f(x) ) | the right-hand limit (approaching a from above) |
| ( \lim_{x \to a} f(x) ) | the two-sided limit (exists only when left and right agree) |
What Are the Types of Discontinuity?
There are three core types. Reading them off a graph is faster than testing limits algebraically, so learn the visual signature of each first.
Removable discontinuity (a hole). Both one-sided limits exist and agree, but the function either has no value there or a value sitting off the curve. The two-sided limit exists; the function value does not match it. You could "fix" it by redefining one point.
Jump discontinuity (a step). Both one-sided limits exist but are different. The graph leaps from one height to another. The two-sided limit does not exist.
Infinite discontinuity (an asymptote). At least one one-sided limit runs off to +∞ or −∞. The graph shoots up or down a vertical asymptote.
How Do You Tell If a Function Is Discontinuous?
Run the three-part continuity test at any suspicious point — usually where a denominator hits zero, where a piecewise rule switches, or where an absolute value or floor changes behaviour.
- Check whether f(a) is defined.
- Find the left-hand and right-hand limits.
- Compare. If the limits disagree, it is a jump. If they agree but f(a) differs or is missing, it is removable. If a limit is infinite, it is infinite.
Examples of Discontinuous Functions
These six progress from a single clean break to mixed-sign, fraction-heavy, and real-world phrasing. Re-trace each on paper.
Example 1
Where is f(x)=( \frac{x-2}{x-4} ) discontinuous?
Set the denominator to zero.
( x - 4 = 0 )
( x = 4 )
At ( x = 4 ) the function is undefined and the one-sided limits run to ±∞.
Final answer: infinite discontinuity at ( x = 4 ).
Example 2
A student claims g(x)=( \frac{x^2 - 9}{x - 3} ) has an asymptote at ( x = 3 ). Is that right?
Wrong attempt. The denominator is zero at ( x = 3 ), leading to a claim of vertical asymptote.
Why it breaks. Plug a value close to 3, say ( x = 2.99 ): the output is near 5.99, not blowing up. An asymptote would send the value toward infinity.
Correct. Factor first.
( g(x) = \frac{(x-3)(x+3)}{x-3} )
The factor cancels, leaving a hole at ( x=3 ) where the curve would sit at 6.
Final answer: removable discontinuity at ( x=3 ), not infinite.
Example 3
Find the discontinuity of the piecewise function ( h(x) = \begin{cases} x + 1, & x < 1 \ 4, & x = 1 \ x + 1, & x > 1 \end{cases} )**
The limit from both sides is ( \lim_{x \to 1}(x+1) = 2 ), but ( h(1) = 4 ).
The limit exists; the value does not match it.
Final answer: removable discontinuity at ( x=1 ).
Example 4
Classify the discontinuity of ( p(x) = \begin{cases} x^2, & x \leq 2 \ x + 5, & x > 2 \end{cases} )**
Left-hand limit: ( \lim_{x \to 2^-} x^2 = 4 ) . Right-hand limit: ( \lim_{x \to 2^+} (x + 5) = 7 ).
Final answer: jump discontinuity at ( x=2 ).
Example 5
Where is the greatest-integer (floor) function f(x)=⌊x⌋ discontinuous?
The floor function returns the largest integer less than or equal to x. At every integer, it steps up by one.
At ( x = 3 ): left-hand limit is 2, and the right-hand limit is 3.
Final answer: jump discontinuity at every integer.
Example 6
A parking garage charges 4$ for the first hour, then 4$ more for each additional hour. Where is the cost function discontinuous, and why does it matter?
The cost holds flat across each hour, then jumps at every whole hour.
Final answer: jump discontinuities at x=1, 2, 3,….
Why Discontinuous Functions Matter
"Many real quantities change in steps, not smoothly."
Smoothness is the exception in the real world, not the rule. The reason the discontinuous function earns its own name is that the things we most need to model — prices, signals, thresholds — refuse to be smooth.
Tripping Points To Avoid
Mistake 1: Calling every zero denominator an asymptote
Where it slips in: Any rational function where a factor cancels.
The correct way: Factor numerator and denominator first. A factor that cancels leaves a removable hole; a factor that survives gives an infinite discontinuity.
Mistake 2: Assuming "defined at the point" means "continuous there"
The correct way: Check all three conditions. Existence is one of three conditions, not the whole test.
Mistake 3: Mixing up left- and right-hand limits on a jump
The correct way: Compute both one-sided limits. A jump is defined by them existing and disagreeing.
Practice Questions
Work through these, then check your answers below.
- Classify the discontinuity of f(x)=( \frac{1}{x - 2} ).
- Classify the discontinuity of g(x)=( \frac{x^2 - 1}{x - 1} ).
- Where is the floor function ⌊2x⌋ discontinuous near x=1.5?
- Classify the discontinuity of h(x)= ( \begin{cases} x + 2, & x < 0 \ x - 2, & x \geq 0 \end{cases} ) at x=0.
- Is the polynomial f(x)=x^3−4x discontinuous anywhere?.
Answers
Answer to Question 1: Infinite discontinuity at x=2. The denominator is zero and does not cancel.
Answer to Question 2: Removable discontinuity (a hole) at x=1.
Answer to Question 3: Jump discontinuity at x=1.5.
Answer to Question 4: Jump discontinuity at x=0.
Answer to Question 5: No. Every polynomial is continuous on all real numbers.
Conclusion
A discontinuous function is one whose graph has at least one break — a point where the three-part continuity test fails.
The three types are removable (a hole), jump (a step), and infinite (an asymptote); each has a distinct graph signature.
- "Defined at the point" is only one of three continuity conditions; the limit must exist and equal the value too.