We are now ready to state the main result of rough paths theory: the continuity of solutions of differential equations with respect to the driving path.

**Theorem:** *Let . Assume that are -Lipschitz vector fields in . Let such that
with .
Let be the solutions of the equations
There exists a constant depending only on and such that for ,
where is the control
*

The proof will take us some time and will be preceeded by several lemmas. We can however already give the following important corollaries:

**Corollary:** *[Lyon’s continuity theorem] Let . Assume that are -Lipschitz vector fields in . Let such that
with .
Let be the solutions of the equations
There exists a constant depending only on and such that for ,
*

This continuity statement immediately suggests the following basic definition for solutions of differential equation driven by -rough paths.

**Theorem:** *Let . Let be a geometric -rough path over the -rough path . Assume that are -Lipschitz vector fields in with . If is a sequence that converges to in -variation, then the solution of the equation*

converges in -variation to some that does not depend on the choice of the approximating sequence and that we call a solution of the rough differential equation:

The following propositions are easily obtained by a limiting argument:

**Proposition:***[Davie’s estimate for rough differential equations]
Let . Let be a geometric -rough path over the -rough path . Assume that are -Lipschitz vector fields in . Let be the solution of the rough differential equation
There exists a constant depending only on and such that for every ,
*

**Proposition:*** Let . Let be a geometric -rough path over the -rough path . Assume that are -Lipschitz vector fields in . Let be the solution of the rough differential equation
There exists a constant depending only on and such that for every ,
*