Definition: Let be a continuous function on . Then if there is a number, z, such that , then there is some number c between a and b such that .
Similarly, if there is a number, y, such that , then there is some number d between a and b such that .
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.