Suppose $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is $C^{\infty}$ at a (continuous and has continuous partial derivatives of all order). Let $U$ be an open set containing $a$.
Then, $f|_U: U\rightarrow \mathbb{R}^m$ is $C^{\infty}$ at a.
My idea was to note that $f|_U$ is continuous since we are restricting the the subspace, and for sufficiently small $t\rightarrow 0$, the partial derivatives of f are the partial derivatives of $f|_U$.
Is this reasoning correct?
Differentiability is a pointwise question that requires local statement. $f$ is said to be differentiable at $x$ if there exists a linear map $u_x : \mathbb{R}^n\to \mathbb{R}^m$ such that for small $h$, $f(x+h) = f(x) + u_x(h) + o(\|h\|)$. It is $\mathcal{C}^1$ if it is differentiable everywhere (pointwise statement) and if $x \mapsto u_x$ is continuous (local statement again). Recursively, you can define what $\mathcal{C}^n$ and thus $\mathcal{C}^{\infty}$ mean.
So if the definitions are true for $f$ over $\mathbb{R}^n$, they are still true for an open subset $U \subset \mathbb{R}^n$!