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

Considere un camino por el que se mueve un automóvil. Si la trayectoria se describe mediante la curva de una función, la posición del automóvil en cualquier momento se puede entender utilizando el concepto de límite a medida que los valores x se acercan a un punto de la curva.A medida que el automóvil viaja a lo largo de la curva, el movimiento suave, es decir, sin interrupciones ni saltos en el camino, sugiere que la función es continua.Si dos puntos se acercan al mismo punto de la curva desde direcciones opuestas y se encuentran suavemente, esto indica que la función es continua en ese punto.Una curva puede tener una discontinuidad en un punto donde la función no está definida, como cuando se produce la división por cero. A medida que x se acerca a 1, las salidas se acercan a 2. Dado que existe este límite, la discontinuidad es removible y la función se puede redefinir en ese punto para que sea continua.Si los límites de la mano izquierda y la mano derecha en un punto no coinciden, la función tiene una discontinuidad en ese punto.Una curva que se dispara hacia arriba o hacia abajo infinitamente sin tocar una línea vertical muestra una asíntota vertical.Los saltos repentinos en una función por partes marcan los puntos donde se rompe la continuidad. Estas se denominan discontinuidades de salto.

continuidad de una función

Una funci&#243;n es continua en un punto a si se cumplen tres condiciones: la funci&#243;n est&#225; definida en a, el l&#237;mite de la funci&#243;n cuando x tiende a dicho punto a existe y ese l&#237;mite es igual al valor de la funci&#243;n. Matem&#225;ticamente, se escribe as&#237;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><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>Esta definici&#243;n garantiza que el gr&#225;fico de la funci&#243;n no presente interrupciones, huecos ni saltos en ese punto. Las discontinuidades se producen cuando alguna de estas condiciones falla. Existe una discontinuidad removible cuando el l&#237;mite bilateral existe, pero la funci&#243;n no est&#225; definida o est&#225; mal definida en ese punto. Estas discontinuidades a menudo pueden resolverse redefiniendo el valor de la funci&#243;n para que coincida con el l&#237;mite.Una discontinuidad de salto se produce cuando los l&#237;mites izquierdo y derecho en un punto son finitos pero desiguales. Esto da lugar a un cambio repentino o &#8220;salto&#8221; en el gr&#225;fico de la funci&#243;n. Una discontinuidad infinita se produce cuando la funci&#243;n tiende al infinito o al infinito negativo cerca de un punto, lo que indica que la funci&#243;n crece sin l&#237;mite. Un ejemplo adecuado es f(x) = 1/x^2 en x = 0.La continuidad en un intervalo cerrado incluye la continuidad en todos los puntos interiores y la continuidad unilateral en los extremos del mismo.

Continuidad de una Función

Una funci&#243;n es continua en un punto a si se cumplen tres condiciones: la funci&#243;n est&#225; definida en a, el l&#237;mite de la funci&#243;n ...

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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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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.

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Types of Function Discontinuities

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