Riemann-Stieltjes Integral Does not Add Up

Question

Does there exist functions $f,\phi : [a,b]\rightarrow \mathbb{R}$ and $c\in [a,b]$ such that the Riemann-Stieltjes integrals $\int_a^c f\,d\phi$ and $\int_c^b f\,d\phi$ exist but $\int_a^b f\,d\phi$ does not exist?