It is a well-known fact that the sum of two unbounded functions may or may not be bounded. One of the simplest examples to observe this is:
$f(x)=x$, $g(x)=-x$ are unbounded but $f+g$ is bounded and $f-g$ is unbounded. We can easily construct examples of of two unbounded functions whose sum is bounded using the scheme: Let $f(x)$ = (Arbitrary unbounded function-1) and $g(x)$ = (Arbitrary bounded function-2) - (Arbitrary unbounded function-1) then $f+g$ = (Arbitrary bounded function-2)
I am trying to construct an example of two unbounded functions whose sum is bounded but which is NOT constructed from the above scheme; that is, an example of two unbounded functions (preferably continuous functions) $f(x)$ and $g(x)$ whose sum cancels off the effect of unboundedness so that it becomes bounded.
It will always be of your form. Let $h(x)=f(x)+g(x)$. Then $g(x)=h(x)-(h(x)-g(x))$ where $h$ is bounded and $h-g=f$ is not.