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What does it mean for a function $\mathbb{R}^n \to \mathbb{R}^m$ to be smooth? I see this in books, but typically we only talk about smoothness when the target set is $\mathbb{R}$.

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Short answer: $f$ is smooth if and only if all of its components are.

Long answer: Observe that you can define partial derivatives for your vector valued $f$ exactly in the same way as you did for scalar valued functions. Indeed the only operations you used on the target space were sums and multiplication by a scalar. So you can just rephrase the "scalar-valued" definition to adapt it to the present case.

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It usually means a $C^{\infty}$ function, or equivalently, a funtion which has partial derivatives for all order.

Another useful fat is that a function $f:\mathbb{R}^n \to \mathbb{R}^m$ is smooth iff every coordinate function $f_i:\mathbb{R}^n \to \mathbb{R}$ is smooth.

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