2. If a binary function is differentiable at a certain point, then the partial derivatives of the function to x and y must exist at that point.
3. If the partial derivatives of a function to x and y exist in a neighborhood of this point and are continuous at this point, then the function is differentiable at this point.
Let the function y=? F(x), if the change Δ x of the independent variable at point X is related to the corresponding change Δ y of the function Δ y = a× Δ x+ο (Δ x), where a has nothing to do with Δ x, the function f(x) is said to be differentiable at point X, and Δ x is called the differential of the function f(x) at point X, and it is called Dy = a×Δx x0, which is recorded as.
Extended data
Weierstrass function is continuous, but it is not differentiable at any point. ?
What if? Then it is differentiable at X0. At this point it must be continuous. In particular, all differentiable functions must be continuous at any point in their domain. The converse proposition does not hold: continuous functions may not be differentiable. For example, a function with vertices, cusps, or vertical tangents may be continuous, but it is not differentiable at abnormal points.
Most functions used in practice are differentiable at all or almost all points. However, Stefan Banach claimed that differentiable functions are a minority in the set of all functions. This means that differentiable functions are not representative in continuous functions. The first continuous but differentiable function discovered by people is the Wilstrass function.
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