Issue with the "definition" of gradient

[Szepi]

Congrats to putting together this nice resource that I am planning to advertise for my students!
A small thing: Eq. (2.1) and Eq. (2.2) supposed to be alternative definitions of the gradient. The actual definition is Eq. (2.2), Eq.(2.1) matches Eq. (2.2) only if f is sufficiently smooth (partial derivatives are continuous).
A very nice explanation that could perhaps be linked to is here: https://mathinsight.org/differentiability_multivariable_subtleties
I think for the sake of the math savvy students of ML, it'd be nice to fix this otherwise minor issue.
Cheers,
Csaba