Decidability of first-order theory of real closed fields with functions
It's undecidable because we can interpret natural numbers (with addition and multiplication).

For example, let $Ind(f)$ be the formula:
$$f(0)=1 \land \forall x \ \big(f(x)=1 \to f(x+1)=1\big)$$

Now we can define the characteristic function of the set of natural numbers using its universality property as a minimal inductive function:
$$Ind(h) \land \forall f \ \big(Ind(f) \to \forall x \ (f(x)=0 \to h(x)=0)\big)$$

(As Emil wrote in his comment we don't even need the language of RCF, we can pick an arbitrary member of the structure as $0$, define a successor function as an injective function whose range lacks our $0$, and then define natural numbers using them. So the undeciability result applies to the second-order language of any infinite structure.)