Cauchy Equation and Equations of Cauchy Type

The equation \( f(x+y)=f(x)+f(y) \) is called the Cauchy equation. If its domain is \( \mathbb{Q} \), it is well-known that the solution is given by \( f(x)=xf(1) \). That fact is easy to prove using mathematical induction. The next problem is simply the extention of the domain from \( \mathbb{Q} \) to \( \mathbb{R} \). With a relatively easy counter-example we can show that the solution to the Cauchy equation in this case doesn\( \prime \)t have to be \( f(x)=xf(1) \). However there are many additional assumptions that forces the general solution to be of the described form. Namely if a function \( f \) satisfies any of the conditions:

  • monotonicity on some interval of the real line;

  • continuity;

  • boundedness on some interval;

  • positivity on the ray \( x\geq 0 \);

then the general solution to the Cauchy equation \( f:\mathbb{R}\rightarrow S \) has to be \( f(x)=xf(1) \).

The following equations can be easily reduced to the Cauchy equation.

  • All continuous functions \( f:\mathbb{R}\rightarrow(0,+\infty) \) satisfying \( f(x+y)=f(x)f(y) \) are of the form \( f(x)=a^x \). Namely the function \( g(x)=\log f(x) \) is continuous and satisfies the Cauchy equation.

  • All continuous functions \( f: (0,+\infty)\rightarrow\mathbb{R} \) satisfying \( f(xy)=f(x)+f(y) \) are of the form \( f(x)=\log_a x \). Now the function \( g(x)=f(a^x) \) is continuous and satisfies the Cauchy equation.

  • All continuous functions \( f: (0,+\infty)\rightarrow(0,+\infty) \) satisfying \( f(xy)=f(x)f(y) \) are \( f(x)=x^t \), where \( t=\log_a b \) and \( f(a)=b \). Indeed the function \( g(x)=\log f(a^x) \) is continuous and satisfies the Cauchy equation.


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