Let $\pi(x)$ denote the number of primes not exceeding $x$. Is $\pi(x)$ differentiable ?
My attempt: It is well known that $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ where $\Re(s)\geq 2$. Since we can't integrate this by parts without differentiating $\pi(x)$, the answer seems to be yes ?
$\pi(x)$ is a nonconstant integer-valued function, so it necessarily has jumps (specifically, it jumps by $1$ at each prime and is constant between primes). It will therefore not be differentiable at each prime, and will have derivative $0$ everywhere else.