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

Considérons un chemin le long duquel une voiture se déplace. Si la trajectoire est décrite par la courbe d’une fonction, la position de la voiture à tout moment peut être comprise en utilisant le concept de limite lorsque les valeurs x s’approchent d’un point sur la courbe.Au fur et à mesure que la voiture se déplace le long de la courbe, un mouvement fluide, c’est-à-dire pas de pauses ou de sauts dans la trajectoire, suggère que la fonction est continue.Si deux points s’approchent du même point de la courbe dans des directions opposées et se rencontrent en douceur, cela indique que la fonction est continue en ce point.Une courbe peut avoir une discontinuité en un point où la fonction n’est pas définie, par exemple lorsque la division par zéro se produit. Lorsque x se rapproche de 1, les sorties se rapprochent de 2. Puisque cette limite existe, la discontinuité est amovible et la fonction peut être redéfinie à ce point pour la rendre continue.Si les limites de gauche et de droite en un point ne correspondent pas, la fonction présente une discontinuité en ce point.Une courbe qui s’élève vers le haut ou vers le bas à l’infini sans toucher une ligne verticale présente une asymptote verticale.Des sauts soudains dans une fonction par morceaux marquent les points où la continuité se rompt. C’est ce qu’on appelle des discontinuités de saut.

continuité d’une fonction

Une fonction est continue en un point a si trois conditions sont satisfaites : la fonction est d&#233;finie en a, la limite de la fonction lorsque x tend vers a existe, et cette limite est &#233;gale &#224; la valeur de la fonction en a. Math&#233;matiquement, cela s&#8217;&#233;crit :<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>Cette d&#233;finition garantit que le graphe de la fonction ne pr&#233;sente aucune rupture, aucun trou ni aucun saut en ce point. Des discontinuit&#233;s apparaissent lorsque l&#8217;une de ces conditions n&#8217;est pas satisfaite. Une discontinuit&#233; amovible existe lorsque la limite bilat&#233;rale existe, mais que la fonction n&#8217;y est pas d&#233;finie ou y est mal d&#233;finie. De telles discontinuit&#233;s peuvent souvent &#234;tre corrig&#233;es en red&#233;finissant la valeur de la fonction pour qu&#8217;elle co&#239;ncide avec la limite.Une discontinuit&#233; de saut se produit lorsque les limites &#224; gauche et &#224; droite en un point sont toutes deux finies mais in&#233;gales. Il en r&#233;sulte un changement brusque, ou &#171; saut &#187;, dans le graphe de la fonction. Une discontinuit&#233; infinie se produit lorsque la fonction tend vers l&#8217;infini ou vers moins l&#8217;infini &#224; l&#8217;approche d&#8217;un point, indiquant que la fonction cro&#238;t sans borne. Un exemple classique est f(x) = 1/x^2 en x = 0.La continuit&#233; sur un intervalle ferm&#233; comprend la continuit&#233; en tout point int&#233;rieur et la continuit&#233; unilat&#233;rale aux extr&#233;mit&#233;s de l&#8217;intervalle.

Continuité d’une fonction

Une fonction est continue en un point a si trois conditions sont satisfaites : la fonction est d&#233;finie en a, la limite de la fonction lorsque x tend ...

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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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One-Sided Limits and Limit Failure

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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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