• If $f$ is defined on $[a,\,b]$ and $g$ is a constant function, then $$\int_a^bf\,dg \;=\; 0.$$
• Let $f$ be continuous on $[a,\,b]$ , $a < c < b$ and $g$ the step function defined as $$g(x) = k \quad \mbox{for\;\;} x < c, \quad g(x) = k\!+\!\alpha \quad \mbox{for\;\;} x > c.$$ Then $$\int_a^bf\,dg \;=\; f(c)\cdot\alpha.$$
• Let $f$ be continuous on $[a,\,b]$ , $a < c < b$ and the function $g$ be otherwise continuous but have in $x = c$ a step of magnitude $\alpha$ . Then $g$ is sum of a continuous function $g^*$ and a step function $$h(x) = 0 \quad \mbox{for\;\;} x < c, \quad h(x) = \alpha \quad \mbox{for\;\;} x > c,$$ and one has $$ \int_a^bf\,dg \;=\; \int_a^bf\,d(g^*\!+\!h) \;=\; \int_a^bf\,dg^*+\int_a^bf\,dh \;=\; \int_a^bf\,dg^*+f(c)\cdot\alpha. $$
• Suppose that $g$ can be expressed in the form $g = g^*\!+\!h$ where $g^*$ is continuous and $h$ a step function having an at most denumerable amount of steps $\alpha_i$ in respectively the same points $c_i$ on the interval $[a,\,b]$ as the function $g$ . If $f$ is Riemann-Stieltjes integrable on $[a,\,b]$ , then

  (1)

  
  
• Suppose that $g = g^*\!+\!h$ (as above) has a finite amount of steps $\alpha_i$ in the points $c_i$ of the interval $[a,\,b]$ but $f$ does not have same-sided discontinuities as $g$ in any of those points. Then $f$ is Riemann-Stieltjes integrable on the interval and the equation (1) is true.

Example. Find the value of the Riemann-Stieltjes integral $$I \;:=\; \int_{-3}^6(x\!-\!\lfloor{x}\rfloor)\,dg(x)$$ where the integrand $f$ is the mantissa function and the integrator $g$ defined by