page_title: Classification of discontinuities
section_title: Jump discontinuity
hierarchical_section_title: Classification of discontinuities / Classification / Jump discontinuity
context_page_description: 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;
  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 is not defined.
context_section_description: Consider the function  
  Then, the point x₀ = 1 is a jump discontinuity.
  In this case, a single limit does not exist because the one-sided limits, L⁻ and L⁺, exist and are finite, but are not equal: since, L⁻ ≠ L⁺, the limit L does not exist. Then, x₀ is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function f may have any value at x₀.
caption: The function in example 2, a jump discontinuity
exif: {}