Step Discontinuity Definition
A step discontinuity or jump discontinuity is a discontinuity where the graph steps or jumps from one connected piece of the graph to another. It is a discontinuity where the limits from the left and right both exist but are not equal to each other. A real-valued univariate function f = f(x) has a step discontinuity at a point x0 in its domain provided that limx→xa- f(x) = L1 < ∞, and limx→xa+ f(x) = L1 < ∞. both exist and L1 ≠ L2.
The notion of step discontinuity shouldn't be confused with the rarely-utilized convention whereby the term jump is used to define any sort of functional discontinuity. Though less algebraically-trivial than removable discontinuities, step discontinuities are far less ill-behaved than other types of singularities such as infinite discontinuities. This fact can be seen in a number of scenarios. For instance, in the fact that univariate monotone functions can have at most countably many discontinuities, the worst of which can be step discontinuities. Unsurprisingly, the definition given above can be generalized to include step discontinuities in multivariate real valued functions as well.
Related Definitions
Sources
“Jump Discontinuity.” From Wolfram MathWorld, mathworld.wolfram.com/JumpDiscontinuity.html.
