Limits and Continuity · Challenge problem

Constants $b$ and $c$ are chosen so that $f(x)=\lfloor x\rfloor^2+\bigl(x-\lfloor x\rfloor\bigr)\bigl(b\lfloor x\rfloor+c\bigr)$ is continuous for every real number $x$, where $\lfloor\,\cdot\,\rfloor$ is the greatest integer function. Find $b+c$.