A small confusing part in the definition of initial value problem

Question

Assume $U\subset\mathbb{R}\times\mathbb{R}^{n}=\mathbb{R}^{n+1}$, $U$ is open and $(t_0, \bf{x}$$_0)\in U$. Assume ${\bf f} (= {\bf f}(t,{\bf x})) : U \to \mathbb{R}$ is continuous. Then the following is called an initial value problem, with initial condition:

\begin{align*} \frac{d\bf{x}}{dt} &= {\bf f}(t, {\bf x}),\\ {\bf x}(t_0) &= {\bf x}_0. \end{align*}

My doubt is $\bf{x}$ is a vector, so $\frac{d\bf{x}}{dt} \in \mathbb{R}^{n}$ but ${\bf f}(t, {\bf x}) \in \mathbb{R}$. Am I correct? So how can they be equal?

Thanks for the help in advance.

Answer 1

Sorry to say that you didn't copy it well. It must be ${\bf f}\colon U\to\mathbb R^n$, precisely because of what you say.

Answer 2

It doesn't make sense. I believe that this is a typo; it should instead be ${\bf f} : U \to \mathbb{R}^n$. One reason to believe that this is the case is that the function name is written in bold which is usually reserved for vector-valued functions.