Theorems of calculus for real-closed fields

Question

Theorems such as

1. Intermediate value theorem

2. Rolle's theorem

3. Mean value theorem

4. Positive/negative derivative implies strictly increasing/decreasing

Hold for polynomials over real-closed fields.

If $F$ is an ordered field define $\lim_{x\to u} f(x)$ to be $L$ such that for an interval $(a, b)\ni L$ there is an interval $(c, d)\ni u$ such that $x\in (c, d)\setminus\{u\} \implies f(x)\in (a, b)$. We can then define what it means for $f:F\to F$ to be continuous, or differentiable.

Suppose that $F$ is a real-closed field. Do theorems 1-4 hold in general, where continuity and differentiability are given as sketched above?