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I am reading Mathematical Analysis by Browder and I have a question about a remark made in connection to Definition 11.1. That definition states:

A map $\mathbf f: A \subset \mathbb R^k \rightarrow \mathbb R^n$ is said to be of class $C^r$ if for each $\mathbf p \in A$ there is an open neighbourhood $U$ of $\mathbf p$ in $\mathbb R^k$, and $\mathbf F$ of class $C^r$ defined in $U$, so that $\mathbf f(\mathbf q) = \mathbf F (\mathbf q)$ for all $\mathbf q \in U \cap A$. We say that a map is smooth if it is $C^\infty$.

The remark below this definition then states that:

In general if $\mathbf f$ is smooth on $A \subset \mathbb R^k$, the differential $\mathrm d\mathbf f$ of $\mathbf f$ is not well-defined (except at interior points of $A$), since the extended map $\mathbf F$ is not well-defined.

This part I understand, and the author also gives an example of when the differential is not well-defined. However, the author them remarks:

But, if $A$ is the closure of its interior and $\mathbf f$ is of class $C^r$ on $A$, then $\mathrm d \mathbf f$, and its matrix $\mathbf f'$, are unambiguously defined on $A$.

How can I convince myself that the last statement is true? (I.e. how can I prove it?)

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