For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.
A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.
A function f(x) is continuous at a point a if and only if the following three conditions are satisfied:
- f(a) is defined
- limx→af(x) exists
- limx→af(x) = f(a)
A function is discontinuous at a point a if it fails to be continuous at a. Discontinuities may be classified as removable, jump, or infinite.
Let f and g be continuous functions. Then
- f + g is a continuous function.
- fg is a continuous function.
- f/g is a continuous function, when g(x) ≠ 0
The Intermediate Value Theorem
Suppose f(x) is a continuous function on the interval [a,b] with f(a) ≠ f(b). If N is a number between f(a) and f(b), then there is a point c between a and b such that f(c) = N.
The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints.
We can locate root using Intermediate value theorem. The theorem fails for discontinuous functions.