Does there exists a function defined on $[0,1] \rightarrow \mathbb{R}$ such that it contains only jump discontinuity but it is not piecewise continuous? 

Jump discontinuity at a point $a$ means $lim\_{x \rightarrow a^{-}}f(x)$ and 
$lim\_{x \rightarrow a^{+}}f(x)$ both exists but not equal

piecewise continuous means every finite subinterval only contains a finite number of discontinuous points and they are all jump discontinuity

My first thought is Dirichlet function and but it appears that it is not the function that I am looking for.....