What functions satisfy $f(x)+f(y-x) = g(y)$?

Question

What functions $f$ have the property that, for all $x$ and $y$: $$ f(x) + f(y-x) = g(y) $$ i.e., the sum does not depend on $x$?

Linear functions obviously this property: if $f(x) = ax+b$, then:

$$ f(x) + f(y-x) = ay+2b = g(y). $$

On the other hand, if we look only at differentiable functions, then only linear functions have this property. By taking the derivative of the first equation as a function of $x$:

$$ f'(x) - f'(y-x) = 0 \iff f'(x)=f'(y-x) $$

Since this is true for every $y$, $f'(\cdot)$ must be a constant function, so $f$ must be linear.

Are there non-differentiable functions with this property?

Kavi Rama Murthy · Accepted Answer · 2019-02-01T09:13:35.603

3

Put $y=x$ to get $f(x)+f(0)=g(x)$. Now verify that $h(x)=f(x)-f(0)$ satisfies $h(x)+h(y-x)=h(y)$. This is same thing as $h(x+y)=h(x)+h(y)$. There are discontinuous functions satisfying this equation. A continuous function $f$ satisfies the given property (that is to say the property stated in the question) iff $f(x)=ax+b$ for some constants $a$ and $b$.

edited Feb 01 '19 at 09:13

answered Feb 01 '19 at 08:57

Kavi Rama Murthy

311,013

The solution $f(x)=ax+b$ is for the original equation in the question, not for additive functions. – Kavi Rama Murthy Feb 01 '19 at 09:12
Then say so. – Did Feb 01 '19 at 09:12
This is what I was missing: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation so if $h$ is continuous, then $h(x) = a x$ for some constant $a$, which implies that $f(x) = h(x) + f(0) = a x + b$ for some constants $a,b$. – Erel Segal-Halevi Feb 02 '19 at 20:32

What functions satisfy $f(x)+f(y-x) = g(y)$?

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