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If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists. In other words, the graph of f has a non-vertical tangent line at the point (x 0, f(x 0)). f is said to be differentiable on U if it is differentiable at every point of U.
The absolute value function : +, = | |, which is not differentiable at = has a weak derivative : known as the sign function, and given by () = {>; =; < This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. For example, the definition of v(0) above could be replaced with any ...
The class consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C 1 {\displaystyle C^{1}} function is exactly a function whose derivative exists and is of class C 0 . {\displaystyle C^{0}.}
Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval [0, 1]. To have the smooth transition in the real interval [a, b] with a < b, consider the function
Of course, the Jacobian matrix of the composition g ° f is a product of corresponding Jacobian matrices: J x (g ° f) =J ƒ(x) (g)J x (ƒ). This is a higher-dimensional statement of the chain rule. For real valued functions from R n to R (scalar fields), the Fréchet derivative corresponds to a vector field called the total derivative.
A function: is a local diffeomorphism if, for each point , there exists an open set containing such that the image is open in and |: is a diffeomorphism. A local diffeomorphism is a special case of an immersion f : X → Y {\displaystyle f:X\to Y} .
Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces. A Fréchet space X {\displaystyle X} is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS , [ 1 ] meaning that every Cauchy sequence in X {\displaystyle X ...
Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If f : M → N is a differentiable function from a differentiable manifold M of dimension m to another differentiable manifold N of dimension n, then the differential of f is a mapping df : TM → TN.