Epäjatkumaisuudet

Epäjatkumaisuudet, in mathematics, refers to points where a function is not continuous. A function is considered continuous if its graph can be drawn without lifting the pen from the paper. At points of discontinuity, this is not possible. There are several types of discontinuities.

Removable discontinuities occur when a hole exists in the graph of a function. This can be "removed" by redefining the function at that specific point. Algebraically, this often happens when a factor in the numerator and denominator of a rational function cancels out.

Jump discontinuities happen when the left-hand limit and the right-hand limit of a function at a point exist but are not equal. This creates a "jump" in the graph. Piecewise functions are a common place to find jump discontinuities.

Infinite discontinuities arise when one or both of the one-sided limits of a function at a point tend towards positive or negative infinity. This typically occurs with rational functions where the denominator approaches zero but the numerator does not, leading to vertical asymptotes.

Other types of discontinuities exist, such as oscillating discontinuities, where the function values fluctuate wildly between two values in any neighborhood of the point. Understanding discontinuities is crucial for analyzing function behavior, solving equations, and in calculus for determining differentiability.