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Continuity in intervals

Definition and examples

A function is called continuous in a subset A of the domain D, if it is continuous at all points in A. We are strongly interested about the case where A is an interval, especially a closed and bounded interval: we shall call this kind of intervals compact intervals.

Functions continuous in compact intervals have two very important properties:

  1. The max-min theorem. Suppose a function f is continuous in the closed interval [a,b]. Then there exist real numbers x1 and x2 such that f(x1) is the minimum value, m, and f(x2) is the maximum value, M, of f in the interval.
  2. The intermediate value theorem. Suppose a function f is continuous in the closed interval [a,b]. Then for every real number y satisfying  f(x1)yf(x2), there exists at least one x0 in [a,b] such that f(x0)=y.


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An application of the intermediate value theorem to numerical analysis.

The intermediate value theorem suggests a very simple technique for finding approximations to solutions of equations. This technique is based on repeated application of the theorem. It is sometimes known as the Bisection technique, and is based on the simple observation that a non-zero real number must be positive or negative, but not both.

Suppose that a function is continuous in the compact interval [a,b]. Suppose further that f(a)f(b)<0. Then the equation
  f(x)=0 has at least one solution in the interval [a,b]. Calculate f(c), where c=(a+b)/2 is the midpoint of the interval [a,b]. Exactly one of the following holds:

If the process does not end, then on each application we have halved the length of the interval under consideration, and so we can approximate the solution as much as we want.

Example: Consider the function f(x)=x3-3x-1, in the interval [-1,0].

The picture below presents a graphical sketch of this process and shows that the middle point of the interval gets closer and closer to the desired solution of the equation.

graph of x^3-3x-1

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first published on march 26 2002 - last updated on september 01 2003