Let $f:[0,1]\to[0,1]$ be a continuous function. Define $h:(0,1)\to[0,1]$ such that, $$h(x)=f(x)-\left\lfloor f(x)\right\rfloor$$Is $h$ continuous? Here $\left\lfloor x\right\rfloor$ is the floor function.
This problem arose due to solving another problem in Real Analysis. Intuitively, it seems that $h$ is continuous but I can neither prove or disprove it. Any help is appreciated.