A special case of the Riemann–Stieltjes integral

Question

A special case of the Riemann–Stieltjes integral is found here. The claim is that if $g(x)$ is continuously differentiable over $\mathbb{R}$, then $$\int_{a}^b f(x)\ \mathrm{d}g(x)=\int_a^bf(x)g^{\prime}(x)\mathrm{d}x.$$ Is it that $g(x)$ needs to be continuously differentiable over ALL of $\mathbb{R}$ or just the interval $[a, b]$? For instance, $\log x$ is continuously differentiable for $x>0$, but not all of $\mathbb{R}$, so if $f(x)=x$ and $b>a>0$, we have $$\int_{a}^b x\ \mathrm{d}\log(x)= b\log b- a \log a -b \log b +b +a \log a -a$$ $$=b-a=\int_a^b \mathrm{d}x.$$