In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is smooth, since the absolute value function on the Dedekind real numbers is constructible, and the absolute value is not smooth. In fact, François Dorais showed here that one can construct an absolute value function for any Cauchy complete Archimedean ordered field. (We define an Archimedean ordered field such that all Archimedean ordered fields are subfields of $\mathbb{R}$.)
However, it is consistent to assume an Archimedean ordered field $R \subseteq \mathbb{R}$ for which there are no maximum or minimum functions; this is equivalent to the condition that there is no absolute value function on $R$ since the absolute value, the minimum function, and the maximum function are all interdefinable with each other in any Archimedean ordered field. Thus, the issues with assuming that every function on $\mathbb{R}$ or any Cauchy complete Archiemdean ordered field is smooth do not apply to functions on $R$.
Thus, is it consistent that every function $R \to R$ is a smooth function using the traditional $\epsilon$-$\delta$ definition of smooth function?
