A continuous function is a mathematical function that has no abrupt changes or breaks in its graph. This means that the function can be drawn without lifting your pencil from the paper, and there are no gaps or holes in the graph.
The Intermediate Value Theorem states that if a continuous function f(x) is defined on an interval [a, b], and if k is any number between f(a) and f(b), then there exists at least one number c in the interval [a, b] such that f(c) = k. In other words, the function takes on every value between f(a) and f(b) somewhere on the interval [a, b].
The Extreme Value Theorem states that if a continuous function f(x) is defined on a closed interval [a, b], then f(x) must have a minimum and a maximum value on that interval. In other words, the function must reach its highest and lowest points somewhere on the interval [a, b].
IVT and EVT are related because they both deal with the behavior of continuous functions. IVT guarantees that a continuous function will take on every value between two points, while EVT guarantees that a continuous function will have a maximum and minimum value on a given interval.
Continuous functions are important because they accurately model many real-world phenomena, such as motion, sound, and temperature. They also allow for the use of powerful mathematical tools, such as the Intermediate Value Theorem and the Extreme Value Theorem, which can be used to solve problems and prove the existence of solutions.