1. Theorem: If the function f is differentiable at point P, the directional derivative of f in any direction exists at point P, ...

—— The above is excerpted from & gt Higher Education Press 125, edited by the Mathematics Department of East China Normal University.

2. Below (it seems that the new function of known mapping is unclear? It is proved that "the function f is differentiable at point P" does not necessarily mean that "the partial derivative of the function at point P is continuous".

—— The above is excerpted from & gt Higher Education Press 1 12, edited by the Mathematics Department of East China Normal University.

3. Now suppose that "a binary function has a directional derivative at a point P in any direction, and it can be inferred that the partial derivative of the function at point P is continuous", then from this condition and the point 1,

We can get: "If the function F is differentiable at point P, then the directional derivative of F exists in any direction at point P, and the partial derivative of the function at point P is continuous", that is, "the function F is differentiable at point P" must be derived.

The partial derivative of a number at point P is continuous, which contradicts point 2. Therefore, it cannot be inferred that the binary function has directional derivative in any direction at point P: the partial derivative of the function at point P is continuous.

continue

Answer the second question:

From the above point 2, it can be seen that "the function f is differentiable at point P" does not necessarily lead to "the partial derivative of the function at point P is continuous".

The first question above is more troublesome, and the second question is actually very easy to understand. F is differentiable at p, and it can be deduced that the partial derivative of F exists at p, but the statement that the partial derivative (partial derivative function) is continuous at p is definitely unknown, because.

For the sake of continuity, we need to examine U(p), but now we only know that F has a partial derivative in P, and we don't know whether F has a partial derivative in U(p).

... fortunately, there is not much dust on the two books of mathematical analysis ... it should be "the problem of mathematical analysis!" " ! ! "Right? I haven't seen Shu Gao, I don't think so ... My answer should be that a one-year-old child can understand it, right?