That's true. If a function is continuous, it's (Riemman) integrable, but the converse is not true.

yes, every continuous function is integrable.

yes

Yes. A well-known example is the function defined as:

f(x) =

* 1, if x is rational

* 0, if x is irrational

Since this function has infinitely many discontinuities in any interval (it is discontinuous in any point), it doesn't fulfill the conditions for a Riemann-integrable function. Please note that this function IS Lebesgue-integrable. Its Lebesgue-integral over the interval [0, 1], or in fact over any finite interval, is zero.

A function may have a finite number of discontinuities and still be integrable according to Riemann (i.e., the Riemann integral exists); it may even have a countable infinite number of discontinuities and still be integrable according to Lebesgue. Any function with a finite amount of discontinuities (that satisfies other requirements, such as being bounded) can serve as an example; an example of a specific function would be the function defined as:

f(x) = 1, for x < 10

f(x) = 2, otherwise