Jätkavus

Jätkavus is a term used in mathematics, particularly in calculus and analysis, to describe a property of functions. A function is said to be continuous if there are no breaks, jumps, or holes in its graph. More formally, a function f(x) is continuous at a point c if three conditions are met: the function is defined at c, the limit of the function as x approaches c exists, and the value of the function at c is equal to the limit. If a function is continuous at every point in its domain, it is called a continuous function.

The concept of continuity is fundamental to many areas of mathematics. For example, the Intermediate Value Theorem states that if a function is continuous on a closed interval, then it must take on every value between its minimum and maximum values within that interval. Similarly, the Extreme Value Theorem guarantees that a continuous function on a closed interval will attain both a maximum and a minimum value on that interval.

Discontinuities, the opposite of continuity, can occur in various ways. A removable discontinuity happens when a function has a hole at a specific point, but the limit exists. An infinite discontinuity occurs when the function approaches infinity or negative infinity as x approaches a certain value, often seen in vertical asymptotes. A jump discontinuity takes place when the function has different values approaching from the left and right sides of a point. Understanding continuity and discontinuity is crucial for analyzing the behavior of functions and solving various mathematical problems.