In my textbook, the definition for Riemann Integrability is as follows:
DEFINITION. A function $f$ is said to be Riemann integrable or more simply integrable on a finite closed interval $[a,b]$ if the limit
$$\int_{a}^{b} f(x) \space \text{d}x = \lim_{\text{max}\Delta x_k\rightarrow 0}\sum_{k=1}^{n}f(x_k^*)\Delta x_k$$
exists and does not depend on choice of the partitions or on the points $x_k^*$ in the subintervals.
My teacher says that any function with a discontinuity is not integrable and as I understand it, a function must be continuous over the entire interval as such quoted:
does not depend on choice of the partitions or on the points $x_k^*$ in the subintervals.
Which means if there is a removable discontinuity, the function is not Riemann Integrable or simply Integrable because the value of $\Delta x_k^*$ now depends on the fact that $x_k^*$ does not land on the $x$ coordinate of the removable discontinuity. Should it land on such value of $x$ then $f(x_k^*)$ would be undefined.
However, I found this similar question Riemann integrability; discontinuity. It says that in order for a function to be Riemann integrable, it must have a Lebesgue Measure of $0$.(Some fancy measure a first year calculus student would not know.) However it later says that the number of discontinuities can be uncountable. Which I believe contradicts the definition because a uncountable number of discontinuities means your choices of partitions have been narrowed down if not to absolute none.
My interpretation of this definition could be very wrong, but my confusion lies here...
Is the definition for Riemann Integrability stated above correct at the most rigorous standards or just outright wrong? and how does Riemann Integrability differ from the implied General Integrability?