How to prove that a bounded and Riemann integrable function is continuous

Question

Given a Riemann integrable function $f : [a,b] \rightarrow \mathbb{R}$. Let $F : [a,b] \rightarrow \mathbb{R}$ be defined by

$$F(x) = \int_a^x f(t)dt.$$

I already showed in a previous exercise that there exists an $M > 0$ such that for all $x,y \in [a,b]$ with $x \leq y$:

$$-M(y-x) \leq F(y) - F(x) \leq M(y-x).$$

In the following exercise I have to prove that $F$ is continuous. Can I use the above to prove this or do I need something else?

Answer 1

The result you already showed is called Lipschitz continuity and it is stronger than continuity: for each $\epsilon>0$ choose $\delta=\epsilon/M$.