The limit of the function at a discontinuous point and even in the most discontinuous function is undefined. There are many properties which show a function to be discontinuous but two most important properties include as follows: Some Properties of a Function Being Discontinuous If the function f(a) is not defined, then in this condition the function is said to be discontinuous.Įvery rational number is continuous except for having its denominator 0, as it becomes discontinuous with the denominator being 0. If the limiting value of xa- f(x) and the limiting value of xa+ f(x) exist and have the same value but are not equal to the value of f(a) then the function is said to be discontinuous.Īt least one or more than one of the limits does not exist then also the function is said to be discontinuous. If the limiting value of xa- f(x) and the limiting value of xa+ f(x) exist but are not having the same value, then the function is said to be discontinuous. In the above equation, the function f(x) can be said to be continuous at the particular set of points only when the limiting value of the given function is equal to the value of f(a), and the function is said to be continuous at the point of x =a.īut the function is said to be discontinuous at the point x =a, while having the following conditions: However, the value of the function is not equal to the value of the limit.Ĭontinuity and discontinuity of a function can be explained with the help of the following example:Ī function f(x) can be said as continuous at a particular point of x =a, if When the function is defined and it has a limit at that point. When the function has no limit at the given point. When the function is not defined at the given point. In other words, the discontinuity is a branch cut along the negative real axis of the natural logarithm for complexes.įor the given function y = f(x), the point x=a is a point of discontinuity if the function is not continuous at this point.Ī Function is Called Discontinuous at a Point if any of the Following Situations take Place: ![]() The removable type of discontinuity is the only type of discontinuity which is fixed and can be redefined, while all other types of discontinuities are designated by the fact that the limit does not exist and cannot be redefined further.Ī discontinuity if a function is referred to as the point at which a Mathematical object is discontinuous. The discontinuity of a function can be defined through the set of points, discrete set along with the dense set and even with the complete domain of the function. ![]() ![]() While continuity of a particular function is very important in the study of Mathematics, similarly discontinuity of a function is also very important in Mathematics, its applications and in its functions. From the left, the function has an infinite discontinuity, but from the right, the discontinuity is removable.Discontinuity of a function can be defined as the point at which the continuity of a particular function cannot be defined in its current domain. The function is obviously discontinuous at $$x = 3$$. We should note that the function is right-hand continuous at $$x=0$$ which is why we don't see any jumps, or holes at the endpoint. Note that $$x=0$$ is the left-endpoint of the functions domain: $$[0,\infty)$$, and the function is technically not continuous there because the limit doesn't exist (because $$x$$ can't approach from both sides). \definecolor \sqrt x$$ (see the graph below).
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |