What do you mean, with a limit? Like how the derivative is defined for just 1 variable?

This definition actually uses limits, since the limit of e1 (c.q. e2) for Δx (c.q. Δy) going to 0 has to be 0. How does this definition "lack rigor"?

You may generalize this definition to functions .

Then the functionfwhich is defined on a neighborhood of

isdifferentiableinaif there exists a vector with

and

Note that this is a multi-dimensional limit and m is exactly the gradient offina.

This definition can then also be generalized to functions