Cauchy functional equation – Wikipedia

L’ Cauchy functional equation is one of the simplest functional equations. This is the following equation, unknown f : ℝ → ℝ:

${displaystyle forall x,yin mathbb {R} quad f(x+y)=f(x)+f(y).}$

In other words, the solutions of this equation are exactly the endomorphisms of the group (ℝ, +).

It is easily shown that any solution f is even ℚ-linear, that is to say also check:

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${displaystyle forall rin mathbb {Q} quad forall vin mathbb {R} quad f(rv)=rf(v).}$

But there are an infinity of non-linear solutions. For a solution to be ℝ-linear, therefore either a homothetic of the real vector right, it is enough that it is continuing at a point or monotonous on an interval of non-zero length. It is enough for this to be increased or reduced over an interval of non-zero length, or even only on a lebesgue-mesurable set of non-zero non-zero ^{[ first ]} .

Either f a solution.

• SO, f is an endomorphism of abélien group, that is to say of ℤ-module, therefore
  ${displaystyle forall nin mathbb {Z} quad forall vin mathbb {R} quad f(nv)=nf(v).}$

  

• We deduce that f is ℚ-linear, that is to say check (in addition to additivity):
  ${displaystyle forall rin mathbb {Q} quad forall vin mathbb {R} quad f(rv)=rf(v).}$

  

  Indeed, all rational r is of the form ^p ⁄ _q with p And q whole and q non -zero, which allows you to write: q f ( rv ) = f ( qrv ) = f ( PV ) = p f ( in ) , SO f ( rv ) = p / q f ( in ) = r f ( in ) .

• A solution f is ℝ-linear if it checks:
  ${displaystyle forall xin mathbb {R} quad forall vin mathbb {R} quad f(xv)=xf(v).}$

  

  ℝ-linear solutions are therefore homotheties, that is to say the applications of the form x ↦ ax (with, necessarily, a = f (first) ).
• Any solution f which is not of this form is far from monotonous, because it is pathological in more than one title:
  • His graph is dense ^{[ 2 ] , [ 3 ]} (in ℝ ² ), so that on any open interval not empty, f is not increased (nor reduced); A stronger , f is discontinuous in every way;
  • all borelian image (by f ) non dense ^{[ 3 ]} (in particular: any Borélien on which f is increased or reduced ^{[ first ]} ) is negligible; it follows that | f | is not increased by any measurable function; A stronger , f is not measurable;
  • and f is not injective then its nucleus is dense so f is “strongly darboux”, that is to say that the image of any interval containing at least two points is ℝ ^{[ 4 ]} .
• In contraped, any “sufficiently regular” solution, i.e. not having one of these pathologies, is a homothety. For example if a solution is increased on a significant Borelian (in particular if it is continuous at one point), or even only if its graph is not dense, then it is a homothetic.

The solutions are exactly the ℚ-linear applications of ℝ in ℝ. Given a Hamel base B From ℚ-vector space ℝ (base whose existence is based on the axiom of choice), the application which with any function of ℝ in ℝ associates its restriction to B is therefore a bijection of all solutions in all applications of B in ℝ.

Many functional equations are reduced to that of Cauchy. For example be the equation, of unknown g : ℝ → ℝ:

${displaystyle forall x,yin mathbb {R} quad g(x+y)=g(x)g(y).}$

Null function is an obvious solution. All the others are strictly positive and check:

${displaystyle forall x,yin mathbb {R} quad ln(g(x+y))=ln(g(x)g(y))=ln(g(x))+ln(g(y)).}$

So these are the functions g = and ^f as f Check the functional equation of Cauchy, and those which are continuous are the exponential functions.