18 июн. 2013 г. · The Cauchy functional equation asks about functions f:R→R such that f(x+y)=f(x)+f(y). It is a very well-known functional equation, which appears in various ...

24 авг. 2012 г. · Suppose f(x+y)=f(x)+f(y). If f:R→R and is measurable, then f(x)=cx. This is referred to as Cauchy's functional equation.

12 нояб. 2015 г. · It is well known that Cauchy's functional equation for f:R→R, f(x+y)=f(x)+f(y)∀x,y∈R, admits highly pathological solutions if no further conditions are ...

21 мая 2018 г. · Let x+y=z for some z, then f(z)=f(z−y)+f(z−x). If f(z)=z then f(z)=z−y+z−x=2z−(x+y)=2z−z=z.

8 сент. 2013 г. · All solutions are "Hamel basis" solutions. Any solution of the functional equation is Q-linear. Let H be a Hamel basis.

24 сент. 2020 г. · If f is a Cauchy function and f(xn)=xkf(x)n−k for positive integers k,n, then f is linear.

26 июл. 2013 г. · It is known that the only measurable solutions to the Cauchy functional equation f(x+y)=f(x)+f(y) are the linear ones (x,y∈R). Does the same ...

10 апр. 2021 г. · Let f:R→R satisfy Cauchy's equation. Suppose in addition that there exists some interval [c,d] of real numbers, > where c<d, such that f is ...

7 апр. 2013 г. · It turns out that as soon as the function is even remotely regular (continuous, bounded on an interval, or measurable), on can show that it has to be linear.

3 янв. 2014 г. · I came upon Cauchy functional equation and tried to prove it. Here is what I have done: We are given the condition, f(x+y)=f(x)+f(y).