Monday, 8 July 2019

calculus - How to prove f(x)=ax if f(x+y)=f(x)+f(y) and f is locally integrable




Suppose f(x) is integrable in any bounded interval on R, and it satisfies the equation f(x+y)=f(x)+f(y) on R. How to prove f(x)=ax?


Answer



Integrate the functional equation with respect to x between 0 and 1. The result is the equation
y+1yf(u)du=10f(x)dx+f(y).
The integral on the left side exists and is continuous in y because f is locally integrable. Therefore the right side is also continuous in y; that is, f is continuous! The rest is clear sailing.


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