# 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.