Hello everyone, I was a little confused about something and hope that someone can help me.

Let be open, and if a function is differentiable on then for each there must exist a linear map such that

Then this is true if and only if . So doesn't this mean that if we can differentiate f, it's then differentiable as a linear map? I know that sounds stupid, but I'm trying to think of the result that if the partial derivatives of f are continuous, then f is differentiable in this special case. So is continuity of the derivative not important in the real case?

What about parametric equations, for example where . On wikipedia it just says that the derivative of f is But do the derivatives of only have to exist for f to be differentiable? Does it matter if they're not continuous?

Thanks for any help