Differentiating/integrating a function infinitely many times

Question

Is there any meaning behind infinitely differentiating or integrating a function? Also, sure, this would cause most functions to either collapse to zero, blow up to infinity, oscillate infinitely (e.g., $sin(x)$ and $cos(x)$), or never change, but are there any functions that would converge to some other function that is continuous at least somewhere upon doing this? Moreover, if there is such a function, how would one go about finding a closed-form expression for its infinite derivative/integral?

Answer 1

the differentiating part:

Let $f(x)$ be such a function that for some function, differentiating it infinitely many times yields it. The immediate consequence is that differentiating $f(x)$ one more time gives precisely $f(x)$, that is, $\frac{df}{dx}(x)=f(x)$. An elementary differential equation argument shows that $f(x)$ must be of the form $Ce^x$, where C any constant.

the integrating part:

Since doing indefinite integrals leave an annoying constant, it's not well-defined that you integrate a function 'infinitely'. Please provide further information :)