- 11.15 Theorem. Let $V$ be open in $\mathbf{R}^{n},$ let a $\in V,$ and suppose that $\mathbf{f} : V \rightarrow \mathbf{R}^{m}$ . If all first-order partial derivatives of $\mathbf f$ exist in $V$ and are continuous at $a$, then $\mathbf{f}$ is differentiable at a.
NOTE: These hypotheses are met if $\mathbf f$ is $\mathcal{C}^{1}$ on $V$
The paragraph is from Wade's Introduction to Analysis, p.398
My question: What does $\mathcal{C}^{1}$ mean? Is that complex set?
Well as it states above, it is a set of classes of functions such that they are partial differentiable and all the partial derivatives are continuous. So basically any function that is of class C1 can be used with the chain rule.