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.

And if the definition of differentiability at a point requires to be defined on an open interval of the point, the definition of differentiability on a set can only be stated for sets for which every point …

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 …

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.

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?

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.

It is not that "closed intervals are used for continuity and open intervals for differentiability" (more on this one later). It is that, for Rolle's Theorem (and the Mean Value Theorem), we need …

If $f (x)$ is differentiable then the derivatives from the left and right must be equal at $x=0$. The derivative of $\dfrac {x} {1+x}$ at $x=0$ is $1$. The derivative ...

Necessary and sufficient conditions for differentiability. Ask Question Asked 12 years, 7 months ago Modified 12 years, 7 months ago