Tricki

## Revision of Use the continuity method from Sat, 09/05/2009 - 03:02

### Quick description

If you can show that the set of parameters obeying a property is non-empty, open, and closed, and the parameter space is connected, then must be obeyed by all choices of the parameter. Thus, for instance, if one wants to prove a property for all in some interval , it suffices to establish the following three facts:

• A base case for some ;

• (Openness) If is true for some , then is true for all sufficiently close to ;

• (Closedness) If is true for some sequence converging to a limit , then is also true.

### Prerequisites

Point-set topology; partial differential equations

### Example 1

(Solving an ODE in a potential well)