Is $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth functions of dual argument

Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument?
Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs based only on Taylor series. That means it is okay to use the formula for argument values where function is analytic (in real numbers). But what about other cases?

1. Does a proof of $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ for non-analytic smooth functions exist?

2. What about dual values $b\epsilon g(a)$ that correspond to argument $a$ where function is non-analytic?