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Use the continuity method

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Quick description

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

  • A base case P(t_0) for some t_0 \in I;

  • (Openness) If P(t) is true for some t \in I, then P(t') is true for all t' \in I sufficiently close to t;

  • (Closedness) If P(t_n) is true for some sequence t_n\in I converging to a limit t \in I, then P(t) is also true.

Prerequisites

Point-set topology; partial differential equations

Example 1

Problem: (Analytic continuation) Show that a real-analytic function f(x) that vanishes to infinite order at one point x_0 \in \R, is identically zero.

Solution: Let P(x) denote the assertion that f vanishes to infinite order at x. Then, by definition of real analyticity, if P(x) holds for some x, then f vanishes within the radius of convergence of the power series at x, and so P(y) must then hold for all y in a neighbourhood of x. On the other hand, since all the derivatives of f are continuous, if P(x_n) holds for a sequence x_n converging to a limit x, then P(x) also holds. Thus the set of x where P(x) is true is non-empty, open, and closed, and is hence all of \R.

Example 2

(Solving an ODE in a potential well)

General discussion

This method can be viewed as a continuous analogue of mathematical induction (or conversely, induction can be viewed as a discrete analogue of the continuity method).

See also A non-trivial circular argument can often be usefully perturbed to a non-circular one.