Lagrange mean value theorem
Suppose is a function defined on a closed interval (with ) such that the following two conditions hold:
- is a continuous function on the closed interval (i.e., it is right continuous at , left continuous at , and two-sided continuous at all points in the open interval ).
- is a differentiable function on the open interval , i.e., the derivative exists at all points in . Note that we do not require the derivative of to be a continuous function.
Then, there exists in the open interval such that the derivative of at equals the difference quotient . More explicitly:
Note that the theorem simply guarantees the existence of , and does not give a formula for finding such a (which may or may not be unique).