I have the function $f$ that is Riemann-integrable in $[a,b]$. First, I need to prove that the function $F:[a,b] \rightarrow \mathbb{R}$ defined by $F(x)=\int_{a}^{x} f(x) dx$ is a Lipschitz function in $[a,b]$.
Second, supposing only that $f$ is bounded in $[a,b]$ (not necessarily Riemann-integrable anymore). Define $H(x)=\overline{\int_{a}^{x}} f(x) dx$. Prove H is a Lipschitz function in $[a,b]$.
If $f$ is Riemann-integrable, then it is bounded. That is, there exists $c>0$ with $|f(x)|\leq c$ for all $x$. Then, for $y\geq x$, $$ |F(y)-F(x)|=\left|\int_x^y f(x)\,dx\right|\leq\int_x^y|f(x)|\,dx\leq\int_x^y c\,dx=c(y-x). $$ For the case where $f$ is bounded, you play the same game as above, but now on the upper sums.