- #1

- 679

- 30

Is this an abuse of Rolle's theorem?

Rolle's theorem

If a real-valued function

*f*is continuous on a proper closed interval [

*a*,

*b*], differentiable on the open interval (

*a*,

*b*), and

*f*(

*a*) =

*f*(

*b*), then there exists at least one

*c*in the open interval (

*a*,

*b*) such that

*f'*(

*c*) = 0

*.*

##[x_1, x_1]##, which means ##x_1\leq x\leq x_1##, is not an interval but a point. And ##(x_1, x_1)##, which means ##x_1<x<x_1##, doesn't make sense. So how can we apply Rolle's theorem?