in this page they are descrbing the forthmentioned technique. and im quite puzzled, because if i get this interval: [1,2] let's take the equation f(x)=x^2. now if we follow the algorithm we find that there might not be a root between them, which is ofcourse absurd. (sqrt2 and sqrt3 are ofcourse included inbetween). anyway, here is the page http://spiff.rit.edu/classes/phys317/lectures/closed_root/closed_root.html
'root' means that f(x) = 0 clearly x^2 is not 0 in the interval you mention, so the algorithm is correct given that everything in the interval [1,2] is the square root of *something*, even your logic there is flawed.
The roots or zeros of a function are synonymous with the "x-intercepts" of that function. The article doesn't mention that the function has to be one-to-one or either increasing or decreasing within the interval for the algorithm to work properly. (Consider a parabola with a vertex below the origin between an interval [x1, x2]. If x2-x1 is greater than the distance between the roots, the algorithm doesn't work.)
The linked algorithm is poorly written. Like signs at the end points of the test interval indicate either no roots or an EVEN NUMBER of roots on the interval. Like wise a sign change on the interval means an odd number of roots on the test interval. The user of the algorithm must have sufficient knowledge of the function to pick a valid starting interval. The best way to get the need information is to plot the function.