Name: types of function discontinuities
Uploaded: 2025-11-26T05:46:05.000Z
Duration: 1 min 23 s
Channel: JoVE

Consider a path along which a car moves. If the path is described by the curve of a function, the car&#8217;s position at any moment can be understood using the concept of a limit as the x-values approach a point on the curve.As the car travels along the curve, smooth movement&#8212;meaning no breaks or jumps in path&#8212;suggests the function is continuous.If two points approach the same point on the curve from opposite directions and meet smoothly, this indicates that the function is continuous at that point.A curve can have a discontinuity at a point where the function is undefined, such as when division by zero occurs. As x gets closer to 1, the outputs approach 2. Since this limit exists, the discontinuity is removable, and the function can be redefined at that point to make it continuous.If the left-hand and right-hand limits at a point don&#8217;t match, the function has a discontinuity at that point.A curve that shoots upward or downward infinitely without touching a vertical line shows a vertical asymptote.Sudden jumps in a piecewise function mark points where continuity breaks. These are called jump discontinuities.

continuïteit van een functie

Een functie is continu in a als aan drie voorwaarden is voldaan: de functie is gedefinieerd in a, de limiet van de functie als x naar a nadert bestaat, en deze limiet gelijk is aan de functiewaarde in a. Wiskundig wordt dit geschreven als<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><img style='aspect-ratio:300/12;' src='https://cloudfront.jove.com/files/ftp_upload/40106/40106_Equation_1_editor_40106.svg' alt='40106_Equation_1.svg' /></figure>Deze definitie garandeert dat de grafiek van de functie op dat punt geen breuken, gaten of sprongen vertoont. Discontinu&#239;teiten treden op wanneer aan een van deze voorwaarden niet wordt voldaan. Een verwijderbare discontinu&#239;teit doet zich voor wanneer de tweezijdige limiet bestaat, maar de functie op dat punt niet of onjuist is gedefinieerd. Dergelijke discontinu&#239;teiten kunnen vaak worden opgeheven door de functiewaarde te herdefini&#235;ren zodat deze overeenkomt met de limietwaarde.Een sprongdiscontinu&#239;teit treedt op wanneer de linker- en rechterlimiet in een punt beide eindig maar ongelijk zijn. Dit resulteert in een plotselinge verandering of &#8220;sprong&#8221; in de grafiek van de functie. Een oneindige discontinu&#239;teit treedt op wanneer de functie naar +&#8734; of &#8722;&#8734; nadert in de nabijheid van een punt, wat aangeeft dat de functie onbegrensd groeit. Een passend voorbeeld is f(x) = 1/x^2 bij x = 0.Continu&#239;teit op een gesloten interval omvat continu&#239;teit in alle binnenpunten en eenzijdige continu&#239;teit aan de eindpunten van het interval.

Continuïteit van een functie

Een functie is continu in a als aan drie voorwaarden is voldaan: de functie is gedefinieerd in a, de limiet van de functie als x naar a nadert bestaat, en ...

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Continuity of a Function

types of function discontinuities

A continuous function has no breaks, holes, or jumps at a point. For a function to be continuous at a point a, three conditions must be met. The function must be defined at a, the limit as x approaches a must exist, and that limit must equal the function’s value.

When any of these conditions fail, the function is discontinuous. One common case is a removable discontinuity. In this case, the two-sided limit exists, but the function is either undefined at the point or given the wrong value. This kind of gap can often be fixed by redefining the function so its value matches the limit.

Another type is a jump discontinuity. Here, the left-hand limit and right-hand limit both exist and are finite, but they are not equal. The graph makes a sudden jump at that point. An infinite discontinuity happens when the function approaches infinity or negative infinity near a point. This means the function grows without bound. A typical example is f(x) = 1/x^2 at x = 0.

Continuity can also be checked on a closed interval. In that case, the function must be continuous at every interior point. The endpoints need one-sided continuity instead of two-sided continuity.

Types of Function Discontinuities

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Secant Lines Approaching a Tangent

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One-Sided Limits and Limit Failure

Limits describe how a function behaves as its input gets close to a target value. They are useful when the function is undefined at that point. Instead of focusing on the exact value, limits show the trend near a critical point and can reveal a discontinuity. One-sided limits look at approach from a single direction. A left-hand limit checks values coming from the left, while a right-hand limit checks values coming from the right. If a function changes differently from each side, the two...

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Vertical Asymptotes and Infinite Limits

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limits with oscillating discontinuities

Oscillating Limits Near Discontinuities

Oscillating discontinuities happen when a function keeps changing value as the input gets close to a point. The output does not settle down to one number. Because of that, the limit at that point does not exist. This kind of discontinuity is different from a jump discontinuity and an infinite discontinuity. In a jump discontinuity, the function shifts suddenly between two values. In an infinite discontinuity, the function grows without bound. A well-known example comes from the function...

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continuity of a function

A continuous function has no breaks, holes, or jumps at a point. For a function to be continuous at a point a, three conditions must be met. The function must be defined at a, the limit as x approaches a must exist, and that limit must equal the function’s value. When any of these conditions fail, the function is discontinuous. One common case is a removable discontinuity. In this case, the two-sided limit exists, but the function is either undefined at the point or given the wrong value. This...

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Horizontal Asymptotes in Function Graphs

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key terms and concepts

Learn how continuous functions work and identify removable, jump, and infinite discontinuities, including one-sided continuity on closed intervals.

Types of Function Discontinuities | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Types of Function Discontinuities

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