কখন নির্দিষ্ট যোগজের অস্তিত্ব থাকে?

For a function to be differentiable, it has to be continuous. Did you know a function doesn't have to be continuous to be integrable?

In a very non-rigorous way, for a definite integral to exist over an interval, you have to be able to draw an area between the curve and the $x$-axis. For example, the function can't have any non-infinitesimal gaps on the interval, or else then there is no area. You can have the area of step functions and other discontinuous functions in such a fashion. Furthermore, if you have a finite number of discontinuities on the interval, then the definite integral always exists.

Another thing we have are improper integrals, which will probably be covered in future articles. It is slightly different from definite integrals not existing.

Our standard rigorous definition of integrability is Riemann integrability.