# Examples of Riemann integrable functions given by Metzler Theorem.

For context, I have to write, for my Calculus II class, a text in which I must explain some theorem, giving a proof of it and some comments that relate the theorem to the rest of the results of the course.

The theorem I've been assigned is what we've called 'Metzler Integrability Theorem' which states the following.

$$\text{Theorem:}$$ Let $$f$$ be a bounded function on the interval $$[a, b]$$. If $$f$$ has a right-hand limit at each point of $$[a, b)$$, then $$f$$ is Riemann-integrable on $$[a, b]$$.

Now, the thing is that I would like to give some examples of functions which we can say are Riemann-integrable thanks to this criterion, but that do not fall in some other category of functions that we already know are Riemann-integrable, such as continuous or monotone functions.

So far the only example I've find is the so called Thomae's function which if I'm not wrong has a limit at each point of its domain, this limit being $$0$$.

To summarize, I would like examples of bounded functions defined on a closed interval that have right-hand limit at each point of its domain that are nor continuous nor monotone. (Ignoring functions $$f:[a, b]\to\mathbb{R}$$ such that, for example, $$f|_{[a, c]}$$ and $$f|_{[c, b]}$$ are both continuous for some $$c\in(a, b)$$).

It is well known that there exists a continuous function $$f$$ on $$[0,1]$$ which is not differentiable at any point. Also, there exists an increasing right-continuous function $$g$$ which is dis-continuous at every rational number. You can now show that $$f+g$$ is right continuous at every point and it is neither monotone nor continuous on any sub-interval of $$[0,1]$$. [A proof of this requires the fact that monotone functions are differentiable at almost all points].