Are Riemann integrable functions on a closed and bounded interval continuous? I know continuous function and monotone functions are Riemann integrable, but I’m not sure if Riemann integrable functions are continuous? 

No, not at all. Here is the precise relation of continuity and Riemann-integrability:

Let $f: [a,b] \to \mathbb{R}$ be a bounded function.
Then $f$ is Riemann-integrable if and only if the set of points at
  which $f$ is discontinuous has Lebesgue-measure $0$.

In particular, since countable sets have measure zero, any function that is discontinuous on a countable set and continuous elsewhere will be Riemann-integrable.
So for example the function
$$ \operatorname{sgn}(x): [-1,1] \to \mathbb{R}$$
will be Riemann-integrable, but this is not continuous at $0$.
