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A function that is Fréchet differentiable for any point of is said to be C 1 if the function : (,); is continuous ((,) denotes the space of all bounded linear operators from to ). Note that this is not the same as requiring that the map D f ( x ) : V → W {\displaystyle Df(x):V\to W} be continuous for each value of x {\displaystyle x} (which ...
If a continuous function on an open interval (,) satisfies the equality () =for all compactly supported smooth functions on (,), then is identically zero. [1] [2]Here "smooth" may be interpreted as "infinitely differentiable", [1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", [2] since these weaker statements may be ...
One thinks of δF/δρ as the gradient of F at the point ρ, so the value δF/δρ(x) measures how much the functional F will change if the function ρ is changed at the point x. Hence the formula ∫ δ F δ ρ ( x ) ϕ ( x ) d x {\displaystyle \int {\frac {\delta F}{\delta \rho }}(x)\phi (x)\;dx} is regarded as the directional derivative at ...
If is differentiable on an open interval and if ′ is a continuous function on , then is called a C 1 function. More generally, f {\displaystyle f} is called a C k function if its derivative f ′ {\displaystyle f'} is C k-1 function.
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 ...
The function : with () = for and () = is differentiable. However, this function is not continuously differentiable. A smooth function that is not analytic. The function = {, < is continuous, but not differentiable at x = 0, so it is of class C 0, but not of class C 1.
Hartogs's extension theorem (1906); [19] Let f be a holomorphic function on a set G \ K, where G is a bounded (surrounded by a rectifiable closed Jordan curve) domain [note 8] on (n ≥ 2) and K is a compact subset of G.
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.