Intermediate Value Theorem

Definition: Let f(x) be a continuous function on [a, b]. Then if there is a number, z, such that f(a) < z < f(b), then there is some number c between a and b such that f(c) = z.

Similarly, if there is a number, y, such that f(a) > y > f(b), then there is some number d between a and b such that  f(d) = y.

Recall from algebra that a continuous function is one you can draw without lifting your pen from the paper. In other words, it doesn’t jump around. This makes sense: since the function must rise from f(a) to f(b), it has to pass through z at some point, or it would have to jump over it (and not be continuous as a result).

Still need help using Riemann Sums? Download Yup and get help from an expert math tutor 24/7.