In mathematical analysis, we usually think that continuous functions are integrable functions, but the reverse is not necessarily the case. In other words, the integrable function is not necessarily continuous.

On the real axis, if the function f(x) is defined on the interval [a, b] and satisfies the Riemann integrable condition, it is said that f(x) is integrable on [a, b]. On the other hand, a continuous function means that if the left and right limits of a function f(x) exist and are equal at a certain point x0, then f(x) is continuous at x0.

Consider the function f (x) = {1, where x is a rational number; 0, x is irrational. We can prove that this function is integrable in any interval [a, b]. However, this function is discontinuous at any point x, because for any x0, we can find a rational number and an irrational number sequence, so that they are both close to x0, but the function values are not equal.

The definition of integrable function is only related to the value of the function in the interval, but has nothing to do with the value of the function at a certain point. Therefore, the discontinuity of a function at one point does not affect its integrability at other points. On the contrary, a function is continuous at a certain point, which does not guarantee that its function value at that point is integrable.

Mathematically, the integrable function is a function with integral. Unless otherwise specified, general integral refers to Lebesgue integral; Otherwise, the function is called "Riemann integrable" (that is, Riemann integral exists), or "Henstock-Kurzweil integrable", and so on.

Riemann integral has achieved great success in the application field, but the application scope of Riemann integral is limited because of its definition. Lebesgue integral is based on Lebesgue measure theory. Functions can be defined on more general point sets. More importantly, it provides a more extensive and effective convergence theorem than Riemann integral. Therefore, Lebesgue integral has a wider application range.