While the conventional "existence of a limit" in 1D is nice (and as a nice side effect gives us the derivative expression too), the general definition is more powerful and it is useful to contrast …

Differentiability refers to the ability to take the derivative of a function and differentiation is the act of taking the derivative.

Differentiability means that the function has a derivative at a point. Continuity means that the limit from both sides of a value is equal to the function's value at that point.

How does one prove differentiability? Ask Question Asked 10 years ago Modified 10 years ago

The reason Rolle's theorem talks about differentiabilty on the open interval $ (a,b)$ is that it is a weaker assumption than requiring differentiability on $ [a,b]$. Normally, theorems might try to …

Sep 23, 2020 · All partial derivatives are directional derivatives in some coordinates, and differentiability is maintained under linear maps. (Note I'm awkwardly trying to use what you …

Jul 20, 2019 · Thanks for this answer. Would you happen to have a resource that would help me practise using this definition? and proving differentiability of functions of two variables?

Why does differentiability imply continuity, but continuity does not mean differentiability? I am more interested in the part about a continuous function not being differentiable. All I could find

Jan 18, 2017 · An excellent reference for the properties of Lipschitz functions is Lectures on Lipschitz Analysis by Juha Heinonen. The differentiability a.e. is Theorem 3.2, page 19. I state …

Dec 11, 2024 · I think it is the definition of Gateaux differentiability simplified for two variable functions but I'm not sure. Also, the continuity of partial derivatives implies differentiability but …