Theorem: If f is continuous for x between a and b and if f (a) and f(b) have opposite signs, then there exists at least one real root of f (x) = 0 between a and b.
One drawback of the bisection method is that it applies only to roots of f about which f (x) changes sign. In particular, double roots can be overlooked; one should be careful to examine f(x) in any range where it is small, so that repeated roots about which f (x) does not change sign are otherwise evaluated (for example, see Steps 9 and 10). Of course, such a close examination also avoids another nearby root being overlooked.
Finally, note that bisection is rather slow; after n iterations the interval containing the root is of length . However, provided values of f can be generated readily, as when a computer is used, the rather large number of iterations which can be involved in the application of bisection is of relatively little consequence.
We can consider f(x) = 3x - , which changes sign in the interval 0.25 < x < 0.27: one may tabulate (working to 4D ) as follows:
(The student should ascertain graphically that there is just one root.)
Let us denote the lower and upper endpoints of the interval bracketing the root at the n-th iteration by and , respectively (with = 0.25 and = 0.27). Then the approximation to the root at the n-th iteration is given by . Since the root is either in  or  and both intervals are of length /2, we see that will be accurate to three decimal places when /2 < 5*. Proceeding to bisection:
(Note that the values in the table are displayed to only 4D.) Hence the root accurate to three decimal places is 0.258.
correct to two decimal places (2D ).