Step Discontinuity — Definition, Graph & Examples

Q: What is the difference between a step discontinuity and a removable discontinuity?

A step (jump) discontinuity has two different finite one-sided limits, so the gap cannot be 'fixed' by redefining a single point. A removable discontinuity has equal one-sided limits (or equivalently, the overall limit exists), but the function is either undefined at that point or defined to the wrong value. You can remove a removable discontinuity by redefining f(c) to equal the limit; you cannot do this for a step discontinuity.

Q: Is a step discontinuity the same as a jump discontinuity?

Yes, the two names mean exactly the same thing. 'Jump discontinuity' is the more common term in most textbooks, while 'step discontinuity' is an equally valid synonym. Both refer to a point where the left-hand and right-hand limits exist but differ.

Q: Can a function have infinitely many step discontinuities?

Yes. The floor function ⌊x⌋ has a step discontinuity at every integer, giving it infinitely many. Other examples include the ceiling function and certain periodic piecewise functions. As long as both one-sided limits exist and differ at each point, each one counts as a step discontinuity.

Step Discontinuity
Jump Discontinuity

A discontinuity for which the graph steps or jumps from one connected piece of the graph to another. Formally, it is a discontinuity for which the limits from the left and right both exist but are not equal to each other.

Graph showing a step discontinuity on x-y axes: curve jumps from an open circle (left piece) to a filled dot (right piece).

Key Formula

\lim_{x \to c^-} f(x) = L \neq R = \lim_{x \to c^+} f(x)

Where:

$c$ = The x-value where the discontinuity occurs
$L$ = The left-hand limit — the value f(x) approaches as x approaches c from the left
$R$ = The right-hand limit — the value f(x) approaches as x approaches c from the right
$f(x)$ = The function being analyzed

Worked Example

Problem: Determine whether the piecewise function has a step discontinuity at x = 2: f(x) = { 3x + 1 if x < 2, and 2x − 3 if x ≥ 2 }

Step 1: Find the left-hand limit by evaluating the piece of the function that applies when x < 2.

\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (3x + 1) = 3(2) + 1 = 7

Step 2: Find the right-hand limit by evaluating the piece of the function that applies when x ≥ 2.

\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x - 3) = 2(2) - 3 = 1

Step 3: Compare the two one-sided limits. Both limits exist, but they are not equal.

L = 7 \neq 1 = R

Step 4: Since the left-hand and right-hand limits both exist but differ, f has a step (jump) discontinuity at x = 2. The size of the jump is |7 − 1| = 6.

\text{Jump size} = |L - R| = |7 - 1| = 6

Answer: Yes, f(x) has a step discontinuity at x = 2. The graph jumps from a value of 7 (approaching from the left) down to a value of 1 (approaching from the right), a jump of 6 units.

Another Example

This example uses a well-known built-in function (the floor/greatest integer function) rather than a custom piecewise definition, showing that step discontinuities arise naturally in standard functions.

Problem: Determine whether the floor function f(x) = ⌊x⌋ (the greatest integer less than or equal to x) has a step discontinuity at x = 3.

Step 1: Recall that the floor function rounds down to the nearest integer. For values just below 3, ⌊x⌋ = 2. For values at or just above 3, ⌊x⌋ = 3.

Step 2: Compute the left-hand limit. As x approaches 3 from the left (e.g., x = 2.9, 2.99, 2.999), the floor value is always 2.

\lim_{x \to 3^-} \lfloor x \rfloor = 2

Step 3: Compute the right-hand limit. As x approaches 3 from the right (e.g., x = 3.1, 3.01, 3.001), the floor value is always 3.

\lim_{x \to 3^+} \lfloor x \rfloor = 3

Step 4: Both one-sided limits exist, but they are not equal, so the floor function has a step discontinuity at x = 3 with a jump of 1.

2 \neq 3 \implies \text{step discontinuity with jump size } 1

Answer: Yes, ⌊x⌋ has a step discontinuity at x = 3 (and indeed at every integer). The graph jumps up by 1 unit at each integer value.

Frequently Asked Questions

What is the difference between a step discontinuity and a removable discontinuity?

A step (jump) discontinuity has two different finite one-sided limits, so the gap cannot be 'fixed' by redefining a single point. A removable discontinuity has equal one-sided limits (or equivalently, the overall limit exists), but the function is either undefined at that point or defined to the wrong value. You can remove a removable discontinuity by redefining f(c) to equal the limit; you cannot do this for a step discontinuity.

Is a step discontinuity the same as a jump discontinuity?

Yes, the two names mean exactly the same thing. 'Jump discontinuity' is the more common term in most textbooks, while 'step discontinuity' is an equally valid synonym. Both refer to a point where the left-hand and right-hand limits exist but differ.

Can a function have infinitely many step discontinuities?

Yes. The floor function ⌊x⌋ has a step discontinuity at every integer, giving it infinitely many. Other examples include the ceiling function and certain periodic piecewise functions. As long as both one-sided limits exist and differ at each point, each one counts as a step discontinuity.

Step (Jump) Discontinuity vs. Removable Discontinuity

	Step (Jump) Discontinuity	Removable Discontinuity
Left-hand limit vs. right-hand limit	Both exist but are NOT equal	Both exist and ARE equal
Overall limit at c	Does not exist (because one-sided limits differ)	Exists (equals the common one-sided limit value)
Can it be 'fixed'?	No — redefining f(c) cannot close the gap	Yes — redefine f(c) to equal the limit
Graph appearance	A visible jump or step between two pieces of the graph	A single hole or misplaced point
Example	Floor function at any integer	(x² − 1)/(x − 1) at x = 1

Why It Matters

Step discontinuities appear frequently in precalculus and calculus when you study piecewise functions, limits, and continuity. Identifying them is essential for determining where a function is continuous, which directly affects whether you can apply theorems like the Intermediate Value Theorem or the Fundamental Theorem of Calculus on a given interval. Beyond mathematics classes, step discontinuities model real-world situations like tax brackets, shipping rates, and rounding functions where values change abruptly at threshold points.

Common Mistakes

Mistake: Confusing a step discontinuity with an infinite discontinuity (vertical asymptote). Students sometimes see a 'break' in the graph and assume any break is a jump.

Correction: For a step discontinuity, both one-sided limits must be finite numbers. If either one-sided limit is ±∞, you have an infinite discontinuity (vertical asymptote), not a step discontinuity. Always compute the one-sided limits numerically.

Mistake: Thinking the function value f(c) determines whether the discontinuity is a jump. Students check f(c) instead of the one-sided limits.

Correction: The classification depends entirely on the left-hand and right-hand limits, not on the actual value of f(c) (or whether f(c) is even defined). Two different finite one-sided limits mean it is a step discontinuity, regardless of what f(c) equals.

Related Terms

Discontinuity — General category that includes step discontinuities
Limit from the Left — One of the two limits compared at a step discontinuity
Limit from the Right — The other one-sided limit compared at a step discontinuity
Graph of an Equation or Inequality — Visual representation showing the jump in the graph
Continuous — A function is continuous where it has no discontinuities
Piecewise Function — Common source of step discontinuities at boundary points
Removable Discontinuity — A different type where the limit exists but differs from f(c)