If a function is made up of some standard built-in functions such as $\ln$, $\exp$, $\sin$, $\cos$, $\operatorname{abs}$ and the basic operations $+$, $-$, $\times$, $\div$, is it true that this function is differentiable everywhere on its open definition domain, except if its expression involves the $\operatorname{abs}$ function, and then it is differentiable on both open domains where the value in $\operatorname{abs}$ is positive resp. negative?
Edit: By the way, can we say the same thing for infinite derivability, since all the built-in functions except $\operatorname{abs}$ are $C^{\infty}$?