Convergence of Riemann integrals on shrinking interval of integrations

Question

For a function $f:[0,1]\rightarrow\mathbb{R}$ which is continuous on $[0,1]$ it is pretty easy to prove that, $\forall r>0$
$$ \Delta_n^{1-r}\,\sum_{j=1}^n\left(\int_{t_{j-1,n}}^{t_{j,n}}f(t)^2\,dt\right)^r\to \int_0^1f(t)^{2\,r}\,dt $$ where $t_{j,n}=j/n$ is the equi-spaced partition of $[0,1]$ and $\Delta_n=1/n$ is the distance between two elements of the partition. The proof is straightforward since, assuming $f$ continuous, we can apply the mean value theorem $$ \int_{t_{j-1,n}}^{t_{j,n}}f(t)^2\,dt = f(c_{j,n})^2\,\Delta_n $$ for a $c_j\in(t_{j-1,n},t_{j,n})$ and then the proof continues by observing that $$ 0\leq \inf_{s\in(t_{j-1,n},t_{j,n})}f(s)^{2\,r}\leq f(c_{j,n})^{2\,r}\leq \sup_{s\in(t_{j-1,n},t_{j,n})}f(s)^{2\,r}<\infty $$ and so $$ \Delta_n^{1-r}\sum_{j=1}^n\left(\int_{t_{j-1,n}}^{t_{j,n}}f(s)^2\,ds\right)^r = \sum_{j=1}^n\left(\frac{\int_{t_{j-1,n}}^{t_{j,n}}f(s)^2\,ds}{\Delta_n}\right)^r\,\Delta_n=\sum_{j=1}^nf(c_{j,n})^{2\,r}\,\Delta_n\rightarrow \int_0^1f(s)^{2\,r}\,ds $$ My guess is that the statement could be generalized to discontinuous functions, for example to cadlag functions with a finite (or perhaps countable) number of jumps in the domain. The main problem is that the mean value theorem is no longer valid, so a different approach should be taken. I thought to apply the theorem on all the sub-intervals in which the function is continuous, would that work or am I loosing something fundamental?