Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Classification of discontinuities
Summary
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
The oscillation of a function at a point quantifies these discontinuities as follows:
in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits of the two sides);
in an essential discontinuity, oscillation measures the failure of a limit to exist; the limit is constant.
A special case is if the function diverges to infinity or minus infinity, in which case the oscillation(mathematics)is not defined (in the extended real numbers, this is a removable discontinuity).
For each of the following, consider a real valued function of a real variable defined in a neighborhood of the point at which is discontinuous.
Consider the piecewise function
The point is a removable discontinuity. For this kind of discontinuity:
The one-sided limit from the negative direction:
and the one-sided limit from the positive direction:
at both exist, are finite, and are equal to In other words, since the two one-sided limits exist and are equal, the limit of as approaches exists and is equal to this same value. If the actual value of is not equal to then is called a . This discontinuity can be removed to make continuous at or more precisely, the function
is continuous at
The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.