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.

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 …

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 …

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 …

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 …

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 …

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.

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

Jul 20, 2019 · If I were to define differentiability of a function of two variables I would say: The function f(x, y) is differentiable at the point (a, b) if, and only if, the partial derivatives f1(a, b) …