On the page Partial derivative on Wikipedia, the following formal definition was found:

enter image description here

I am wondering if in this definition, the condition that $U$ being open is always necessary. For example, if in $\mathbb{R}^n$, a function $f$ is defined on the set of union of all axises: $A=\{(x_1,\ldots,x_n)\mid \text{all but one }x_i \text{'s are zero}\}$, it seems to me that one can still compute all partial derivatives of $f$ at $(0,\ldots, 0)$ without requiring the domain of $f$ being an open set.

Should we loosen the condition that $U$ is open to something like "U contains the intersection $\{(x_1,\ldots,x_n)+(a_1,\ldots, a_n)\mid \text{all but one }x_i \text{'s are zero}\}\cap B$, where $B$ is an open ball containing $(a_1,\ldots, a_n)$"? In this case the domain of $f$ is not necessarily an open set but the definition of the partial derivative at $(a_1,\ldots, a_n)$ still makes sense to me. Did I miss anything here?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.