1. Let $g(x) = C$ everywhere. Is true that for any function $f$ $\int\_a^b f dg$ exists?

My idea was to consider the definition, and since for any points $x\_k, x\_{k+1}: g(x\_k)-g(x\_{k-1}) = 0$ RS-sum is always zero for any function f. Am I wrong somewhere? 

2. Let $I= ${$x\_i$}$\_1^\infty$ - countable set of points where and only where $g(x)\neq0$

$$ g\_n(x)=
\begin{cases}
0, x\in I\setminus (x\_1,x\_2,x\_3,\dots,x\_n) \\
g(x), \text{elsewhere}
\end{cases}$$
Then $V([a,b],g\_n) \leq V([a,b], g)$.

Not sure if I have any idea how to use that I is countable.