AP Syllabus focus:
‘Extend the definition of the definite integral to functions with removable or jump discontinuities and determine when the integral still exists.’
Understanding how definite integrals behave when a function is not perfectly continuous is essential, because real-world rate functions often contain isolated gaps or abrupt changes. This section explains when accumulated area is still well-defined.
Integrals with Removable or Jump Discontinuities
Definite integrals remain meaningful for many functions that are not continuous everywhere. In AP Calculus AB, you must recognize when removable and jump discontinuities do not prevent an integral from existing, and how the structure of these discontinuities affects the interpretation of accumulated area.
Types of Discontinuities Relevant to Integrals
A removable discontinuity occurs when a function has a “hole” at a single point but can be redefined to make the function continuous there.
Removable Discontinuity: A point at which a function is undefined or mismatched with its limit, even though the limit exists and is finite.
A jump discontinuity occurs when a function suddenly shifts to a different function value at a point, creating distinct left- and right-hand limits that are not equal.
Jump Discontinuity: A point where the left-hand limit and right-hand limit both exist but are not equal, creating an instantaneous jump in function value.
Single discontinuities typically do not cause issues for integration because definite integrals measure area, which depends on the behavior of the function over intervals, not at isolated points.
This graph illustrates a function with a removable discontinuity: the line is smooth except for a single open circle where the function value is missing. Because this gap occurs at only one -value, it contributes no area to any definite integral . The figure visually supports the idea that integrals can still exist even when a function has isolated “holes.” Source.
A removable discontinuity behaves like a single missing point, while a jump discontinuity behaves like a sudden shift.
This figure shows a function with a jump discontinuity: the graph sits on one horizontal line to the left and abruptly jumps to a different horizontal line to the right. The open circle and filled dot at the same -value emphasize that the left-hand and right-hand limits exist but are unequal. This picture reinforces the idea that a jump discontinuity is a finite step, not an infinite spike, so the integral still focuses on the area under each piece. Source.
A jump discontinuity behaves like a sudden shift. Neither type introduces infinite height or unbounded behavior, which is important for determining integrability.
Why Integrals Can Still Exist with Isolated Discontinuities
The definite integral is defined as the limit of Riemann sums. Riemann sums involve products of function values and small widths. Because a single point has zero width, changing or removing the function value at that point does not affect the total area.
Key Reasons the Integral Still Exists
Points have zero area, so isolated discontinuities do not change accumulated area.
Riemann sums rely on interval behavior, not individual function values.
Boundedness is preserved for typical removable and jump discontinuities, preventing unbounded integrand issues.
These ideas allow the definite integral to be extended naturally to many functions that are not continuous everywhere.
Conditions for Integrability with Discontinuities
For a definite integral to exist on , the integrand must be bounded and have only finitely many removable or jump discontinuities.
A Function Remains Integrable If:
It has a finite number of removable discontinuities.
It has a finite number of jump discontinuities.
It is bounded on the interval.
It has no infinite discontinuities within .
A Function Fails to Be Integrable If:
The discontinuity is infinite, meaning the function grows without bound near the point.
There are infinitely many discontinuities dense on the interval.
The function becomes unbounded on any subinterval.
These criteria ensure that Riemann sums converge to a well-defined accumulated area.
Interpreting Area with Removable Discontinuities
A removable discontinuity does not change the area under a curve. Because the value at a single point contributes nothing to the integral, the definite integral treats the function as though the point were “filled in” with its limiting value.
Features of Removable Discontinuities in Integration
They behave like a tiny missing dot on the graph.
Redefining the point leaves the definite integral unchanged.
The accumulation of area proceeds smoothly across the discontinuity.
This means a function with occasional holes still produces a meaningful total accumulation.
Interpreting Area with Jump Discontinuities
Jump discontinuities create a vertical gap between two distinct function values, but they do not create an infinite barrier.
Key Interpretive Ideas
The integral treats each side of the jump separately but continuously within each interval.
The area “skips” the vertical jump; since the jump has zero width, it contributes no area.
Accumulated change is computed by integrating along the function values on each side of the discontinuity.
In this way, jump discontinuities resemble step changes in rate functions, which are common in applied contexts such as pricing structures or piecewise physical behaviors.
Integrals and the Behavior of Piecewise Functions
Many functions with removable or jump discontinuities are naturally described using piecewise definitions. As long as each piece is continuous on its open interval and only finitely many pieces meet, the definite integral exists.
Important Structural Characteristics
Each piece is continuous except possibly at endpoints.
The boundaries between pieces introduce at most jump discontinuities.
Integrability depends on boundedness, not continuity.
Piecewise rate functions are well-handled within the definition of the definite integral, reinforcing that integrals measure area over intervals, not at specific points.
Practical Interpretations for Accumulated Change
Since integrals represent accumulated change, discontinuities should be interpreted in terms of their effect on total accumulation.
Interpretation Guidelines
Removable discontinuity: No effect on accumulated change.
Jump discontinuity: Instantaneous shift in the rate, but the total accumulation remains well-defined.
Overall accumulation: Determined by integrating steadily on each subinterval where the function is defined.
These interpretations align with the AP focus on using integrals to quantify total change even when the underlying rate behaves irregularly at isolated points.
FAQ
A function can still be integrable if it has multiple removable or jump discontinuities, provided there are only finitely many of them on the interval.
If the number of such discontinuities becomes infinite and they accumulate densely, the function may fail to be integrable, even if each individual discontinuity is mild.
Boundedness across the entire interval remains essential.
They do not directly affect the computed area, but they can influence how the integral is evaluated from a piecewise perspective.
For example, when using analytical or numerical methods, each subinterval between discontinuities may be treated separately, even though the final integral value is unaffected.
This matters more in practical computation than in the theoretical definition.
Look for these visual cues:
• A removable discontinuity appears as a single hole with no vertical gap.
• A jump discontinuity shows two distinct function values at the same x-location.
If the graph contains a vertical asymptote, it is not part of this subsubtopic, as that indicates an infinite discontinuity, which affects integrability differently.
You cannot eliminate a jump discontinuity by redefining the function at a single point.
A jump occurs because the left-hand and right-hand limits disagree, so no single value can bridge the gap.
However, you may define separate piecewise sections to model the function more clearly, even though the jump remains.
Discontinuities do not disrupt Riemann sums because each term reflects the value at a point multiplied by a small width.
In numerical approximations:
• Removable discontinuities simply behave as single missed points.
• Jump discontinuities are handled by sampling either side of the jump as the partition width shrinks.
Computational issues only arise if the discontinuity is infinite or if sampling points repeatedly land exactly at undefined locations.
Discontinuities do not disrupt Riemann sums because each term reflects the value at a point multiplied by a small width.
In numerical approximations:
• Removable discontinuities simply behave as single missed points.
• Jump discontinuities are handled by sampling either side of the jump as the partition width shrinks.
Computational issues only arise if the discontinuity is infinite or if sampling points repeatedly land exactly at undefined locations.
Practice Questions
Question 1 (1–3 marks)
A function g is defined on the interval [0, 4] and is bounded. The graph of g has a single removable discontinuity at x = 2 but is otherwise continuous.
Explain whether the definite integral from 0 to 4 of g(x) dx exists, and justify your answer.
Question 1
• 1 mark: States that the definite integral from 0 to 4 of g(x) dx exists.
• 1 mark: Identifies that a removable discontinuity does not prevent the integral from existing.
• 1 mark: Explains that a single point has no width, so it contributes no area and does not affect the value of the integral.
Question 2 (4–6 marks)
A function f is defined on [−3, 3] and is bounded. It has a jump discontinuity at x = −1 and a removable discontinuity at x = 2. The function is continuous everywhere else.
(a) State whether the definite integral from −3 to 3 of f(x) dx exists.
(b) Explain how the two discontinuities affect (i) the existence of the integral and (ii) the accumulated area.
(c) Suppose f has an additional infinite discontinuity at x = 0. Explain how this affects the integrability of f on [−3, 3].
Question 2
(a) (2 marks)
• 1 mark: States that the definite integral from −3 to 3 of f(x) dx exists.
• 1 mark: Justifies this by stating that a finite number of removable or jump discontinuities does not prevent integrability if the function is bounded.
(b) (2 marks)
• 1 mark: States that the discontinuities do not affect the existence of the integral because they are isolated and the function remains bounded.
• 1 mark: States that the accumulated area is computed as usual, since the discontinuities occur at single x-values that contribute no area.
(c) (2 marks)
• 1 mark: States that an infinite discontinuity at x = 0 prevents the definite integral from being defined in the standard sense.
• 1 mark: Explains that unbounded behaviour near x = 0 means the integral may diverge and therefore may fail to exist as a proper Riemann integral.
