hyppyepäjatkuvuuskohta

Hyppyepäjatkuvuuskohta, meaning "jump discontinuity" in English, is a concept in calculus that describes a specific type of discontinuity in a function. At a jump discontinuity, the function does not have a single, defined value at a particular point. Instead, the function "jumps" from one finite value to another as the input approaches the point from the left and from the right.

More formally, a function f(x) has a jump discontinuity at a point x = c if the left-hand limit and the right-hand limit at that point both exist but are not equal. The left-hand limit, denoted as lim_{x->c^-} f(x), represents the value the function approaches as x gets closer to c from values less than c. Similarly, the right-hand limit, lim_{x->c^+} f(x), represents the value the function approaches as x gets closer to c from values greater than c. For a jump discontinuity, lim_{x->c^-} f(x) != lim_{x->c^+} f(x).

It is important to note that for a jump discontinuity to exist, both of these one-sided limits must be finite. If either limit is infinite, the discontinuity would be classified differently (e.g., an infinite discontinuity). Jump discontinuities are common in piecewise functions, where the definition of the function changes abruptly at a specific point. For example, a step function exhibits jump discontinuities at the points where the steps occur. Understanding jump discontinuities is crucial for analyzing the behavior of functions and for applying various calculus theorems.