Let $g$ be a nonnegative function.
We are given $\sum_{y \leq n \leq x} f(n) = F(x) - F(y) + O(g(x)+ g(y))$.
I am wondering if the following is equivalent, and if so how to prove it:
$\sum_{y \leq n \leq x} f(n) = F(x) - F(y) + O(g(x))+ O(g(y))$.
One thing thats confusing me is if I write $h = O(g(x)+ g(y))$, then should the argument of $h$ be $x$ or $y$ or both $x$ and $y$ as in $h(x,y)$ or a new variable entirely?
In any case $h \leq C(g(x) + g(y))$ for some constant $C$ and all sufficiently large values of the parameter(s) of $h$. This is equivalent to $h \leq Cg(x) + Cg(y) = O(g(x)) + O(g(y)).$
Conversely if $h(x) = O(g(x))$ and $k(y) = O(g(y))$, then $h(x) + k(y) \leq Cg(x) + Dg(y) \leq E(g(x) + g(y))$ where $E \geq C,D$. And so $h(x) + k(y) = O(g(x) + g(y)$.
Is this a correct argument showing $O(g(x)+ g(y))$ the same thing as $O(g(x))+ O(g(y))$?
