Let $sgn:\mathbb{R}\to\mathbb{R}$ be the signum function, Let $f:[-1,1]\to\mathbb{R}$ be continuous. Show that $f$ is Riemann-Stieltjes integrable and $\int_{[-1,1]} f \delta sgn=2f(0)$
Where $sgn(x)=\begin{cases} 1& x>0\\0&x=0\\-1&x<0\end{cases}$
And $sgn[J]$ denotes the sgn length of an interval $J=(a,b)$ as $sgn(b)-sgn(a)$
I think the idea with this is that if $P$ is a partition of $[-1,1]$ then for each interval $J\in P$, the length of $J$ will be $sgn(b)-sgn(a)$ which will be $0$ everywhere but an interval with an endpoint at $0$ and there can be at most $2$ disjoint intervals with endpoints at $0$.
So let $\epsilon>0$ then since $f$ is continuous there exists a $\delta>0$ then if $y\in(-\delta,\delta)$ then $\vert f(y)-f(0)\vert<\epsilon$
Let $P$ be a partition with $\vert J\vert <\delta$
Then I want to define a piecewise constant function $g,h$ such that $g(x):=\sup f(x)$ and $h(x):=\inf f(x)$ for $x\in J$
Then $\overline\int_{[-1,1]}f\,\delta\,sgn\leq \int_{[-1,1]}g\,\delta\,sgn=\sum_{J\in P}\sup_{x\in J} f(x) sgn[J]$
I believe I can assume that there exist $J_k,J_L$ such that the lower bound of $J_k$ is $0$ and the upper bound of $J_L$ is $0$, since if they aren't then I can take the common partition of $P$ and $\{[-1,0),[0,1)\}$
Thus if $J\in P$ then $sgn[J]=0$ for all $J$ except $J_k,J_L$ where $sgn[J_k]=sgn[J_L]=1$
Then $\sum_{J\in P}\sup_{x\in J} sgn[J]=\sup_{x\in J_K} f(x)+\sup_{x\in J_L}f(x)$
Similarly I can prove that $\underline\int_{[-1,1]}f\, \delta\, sgn\leq \underline\int_{[-1,1]}h\, \delta\,sgn=\inf_{x\in J_K} f(x)+\inf_{x\in J_L}f(x)$
Since $\vert J_k\vert=\vert J_L\vert <\delta$ we know that $\vert f(y)-f(0)\vert<\epsilon$ for every $y\in J_k,J_L$ thus $\vert\sup f(y)-f(0)\vert<\epsilon$ and $\vert \inf f(y)-f(0)\vert<\epsilon$ for $y\in J_k,J_L$
Then for $y\in J_k,J_L$ we get $\vert \sup f(y)-\inf f(y)\vert\leq\vert f(y)-f(0)\vert+\vert \inf f(y)-f(0)\vert <2\epsilon$ by triangle inequality
Then $\overline\int_{[-1,1]}f\,\delta\, sgn-\underline\int_{[-1,1]}f\,\delta\, sgn\leq 4\epsilon$
Thus the function is Riemann-Stieltjes integrable and the Riemann integral is $2f(0)$ since the $\sup,\inf$ are epsilon close to $f(0)$ on $2$ intervals.
Does this proof seem correct?