If function is not continuous then it means it's not differentiable and that would mean that it should not be integrable as differentiation is the reverse process of integration.
I am talking about INDEFINITE INTEGRATION.
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$\begingroup$"If function is not continuous then it means it's not differentiable and that would mean that it should not be integrable."
You are confused. The function $f(x)=|x|$ is not (everywhere) differentiable, but it is integrable on any closed interval.
In your case, the function $f(x)=\frac{1}{x}$ is integrable on any closed interval away from zero.
In fact, there are many integrable functions that are not even continuous.
$\endgroup$ $\begingroup$When mathematicians discuss whether a function is integrable, they aren’t talking about the difficulty of computing that integral. Each year, mathematicians find new ways to integrate classes of functions. However, this fact doesn’t mean that previously nonintegrable functions are now integrable.
When mathematicians say that a function is integrable, they mean only that the integral is well defined, that is, that the integral makes mathematical sense.
In practical terms, integrability hinges on continuity: if a function is continuous on a given interval, it’s integrable on that interval. Additionally, if a function has only a finite number of some kinds of discontinuities on an interval, it’s also integrable on that interval.
Many functions, such as those with discontinuities, sharp turns, and vertical slopes, are nondifferentiable. Discontinuous functions are also nondifferentiable. However, functions with sharp turns and vertical slopes are integrable.
For example, the function $y = |x|$ contains a sharp point at $x = 0$, so the function is nondifferentiable at this point. However, the same function is integrable for all values of $x$. This is just one of infinitely many examples of a function that’s integrable but not differentiable in the entire set of real numbers.
So, surprisingly, the set of differentiable functions is actually a subset of the set of integrable functions.
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