I have been doing lots of calculus these days and i want to confirm with you guys my understanding of an important concept of calculus.
Basically, in the initial phase,students assume that integration and differentiation are always associated to each other, i.e., a function which is integrable is also differentiable at the same time. But having explored on it more, i found out its not true at all and does not hold always. Many a times (or should i say infinitely times) a function can be integrable on an interval while its not differentiable on that same interval (and vice versa) .
What i want to ask is this : i recently read a conclusion on the above mentioned concept which is :
{Differentiable functions} $\subset$ {Continuous functions} $\subset$ {Integrable functions}
i.e., each is a proper subset of the next. Now, "Differentiable functions set" is a proper subset of "Continuous functions set"… that is very well understood without a doubt as every continuous function may or may not be differentiable. I have problem with the next relation which is :
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"Continuous functions set" is a proper subset of "Integrable functions set"…Why is this so???
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I am just not able to visualize this. I know that a bounded continuous function on a closed interval is integrable, well and fine, but there are unbounded continuous functions too with domain R , which we cant say will be integrable or not.
So, my question is simple. Why are there more number of elements in the "Integrable functions set" than "Continuous functions set" (here by elements i mean integrable and continuous functions ofcourse) ???. So,this is it… Can anyone plz help me understand this out in as simple words as possible. I know i need some kind of visualization which i guess is easy, but i could not make it out on my own, so i turned to u guys.Thanks for any help.
Best Answer
Let $g(0)=1$ and $g(x)=0$ for all $x\ne 0$. It is straightforward from the definition of the Riemann integral to prove that $g$ is integrable over any interval, however, $g$ is clearly not continuous.
The conditions of continuity and integrability are very different in flavour. Continuity is something that is extremely sensitive to local and small changes. It's enough to change the value of a continuous function at just one point and it is no longer continuous. Integrability on the other hand is a very robust property. If you make finitely many changes to a function that was integrable, then the new function is still integrable and has the same integral. That is why it is very easy to construct integrable functions that are not continuous.