Question

When does partial derivative exist for some function in some point?

Answer 1

Is this when function is continuous in this point?

Answer 2

continous and differentiable

Answer 3

and function is continuous if :
-exist limit of this function in point
-exist value of this funtion in point
-limit is the same as a value?

Answer 4

The partial derivative exists when the function is differentiable in the direction along which you are differentiating. For example, for a function z=f(x,y), ∂z/∂x can exist even if the function is not differentiable in the y direction.

A function is differentiable at a point if and only if the function is continuous at that point and the limit of the derivative from the left and the right approach the same value. This means that for a function to be differentiable it must be continuous (the function exists, the limit exists, and the limit=the function) and it must be "smooth" i.e. it cannot make sharp, angular turns.