Demonstration of the differentiability of a continuous function

[Telemachus]

Homework Statement

I have some doubts about the demonstration of the differentiability. If I'm asked to proof that an average function is differentiable on all of it domain, let's suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I demonstrate that its differentiable? May I use the limit with generic points $$(x_0,y_0)$$? I mean, if I use this limit (the one with the function and the tangent plane over the square root that represents a disk), and its a differentiable function, with this generic points the limit should give zero, right?

Bye there, thanks for posting.

PD: I'm talking for function of two or more variables.

 

[lippyka]

Homework EquationsThe definition of differentiability is that a function f is differentiable at a point (x_0,y_0) if the limit of lim_{(h,k)->(0,0)} (f(x_0 + h, y_0 + k) - f(x_0, y_0))/(h^2 + k^2)^1/2exists and is finite.The Attempt at a SolutionYes, you can use the limit with generic points (x_0,y_0) to demonstrate that an average function is differentiable on all of its domain. As long as the function is continuous on its domain and the limit exists and is finite, then the function is differentiable.