Prove that $f(x)=\lfloor x\rfloor$ is Riemann integrable on $[0,5]$

Question

I have several doubts about this exercise because one of the conditions a function must have to be Riemann integrable is to be continuous in that interval, condition $\lfloor x\rfloor$ does not meet. How is this exercise done? Or what does the approach have to be?

Prove that $f(x)=\lfloor x\rfloor$ is Riemann integrable on [0,5] and calculate $\int_0^5 \lfloor x\rfloor \,dx$, where $\lfloor x\rfloor = floor(x)$

Thanks in advance.

Answer 1

Monotone functions are integrable! To calculate the integral, integrate over [j,j+1] for j=0,1,2,3,4 and add.

Answer 2

Hint: try to prove (or find a prove in your textbook) that a function is Riemann integrable if it has only finitely many discontinuities in the interval of integration.

More generally, a function is Riemann integrable if it has countable many discontinuities. You might also want to try to find a prove for this.

Answer 3

A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure).

https://en.wikipedia.org/wiki/Riemann_integral