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).

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