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 the two. …

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

Differentiability is a stronger condition than continuity. If f f is differentiable at x = a x = a, then f f is continuous at x = a x = a as well. But the reverse need not hold. Continuity of f f at x = a x = a …

In the definition of pointwise differentiability we replaced the condition " x x is contained in an open neighborhood" by " x x is a limit point". And in the definition of differentiability on a set we just …

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 the steps of the …

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 not vice versa.

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.

We wanted to determine whether or not f(x) f (x) is differentiable at 0 0. I already know that f(x) f (x) is continuous at 0 0 using the definition of continuity. If I am correct, to show differentiability we have to …

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?

Necessary and sufficient conditions for differentiability. Ask Question Asked 12 years, 8 months ago Modified 12 years, 8 months ago