I am wondering what operations preserve smoothness of functions. For example, sum of two smooth functions, product of two smooth functions and composition of two smooth functions is also smooth. Is it the case for integrals of smooth functions?
Also, do there exist functions that have well-defined 2nd derivative but not 1st derivative? I would like to delve further into the topic, links to references and sources would be appreciated.
Thanks!
If we talk about "simple" smooth functions, i.e $f:\mathbb R \rightarrow \mathbb R$, then the set of all smooth function form a vector space over the field of real numbers.
So any property of a vector space can be apllied to smooth functions.
As for your specific question, a smooth function on a closed interval $[a,b]$ is continous there, therefore it is integrable. And we know that integrating a function makes it "nicer", or "smoother" so of course the integral operator also preserve the smoothness of a function.
Furthemore, if the function is $C^{\infty}$ (which is usually what we mean when we say smooth), then the derivative operator also preserve the smoothness.
As for your second question - No. A function can't have a second derivative if it does not have a first derivative, because the second derivative of $f$ is simply the derivative of $f'$, but since $f'$ is not defined, then it is not differentiable.