Hyppysepäjatkuvuus

Hyppysepäjatkuvuus, sometimes translated as jump discontinuity or simple discontinuity, refers to a type of discontinuity in a mathematical function where the function has a finite jump from one value to another at a specific point. At a point x = a, a function f(x) exhibits a hyppysepäjatkuvuus if the limit of the function as x approaches a from the left exists and is finite, and the limit of the function as x approaches a from the right also exists and is finite, but these two limits are not equal. Formally, this means that lim_{x->a^-} f(x) = L1 and lim_{x->a^+} f(x) = L2, where L1 and L2 are finite numbers and L1 != L2.

The value of the function at the point x = a, f(a), can either be equal to L1, L2, or a third finite value, or it might be undefined. Regardless of the value of f(a), if the left and right limits exist but are unequal, the discontinuity is classified as a hyppysepäjatkuvuus. This is in contrast to other types of discontinuities such as removable discontinuities, where the left and right limits are equal but the function is either undefined at the point or has a different value, or infinite discontinuities, where at least one of the one-sided limits is infinite. Examples of functions exhibiting hyppysepäjatkuvuus include the Heaviside step function at x=0 and the floor function at integer values.