C1functie

C1functie, or C^1 function, is a term used in calculus and analysis to denote a function that is continuously differentiable. In practice, this means the function is differentiable on its domain and its derivative is a continuous function. The concept applies to real-valued functions of one variable as well as to functions between Euclidean spaces.

In one variable, a function f: I → R on an interval I is C^1 if f is differentiable for every x in I and the derivative f′ is continuous on I. In several variables, a function f: U → R^m, with U an open subset of R^n, is C^1 if all first-order partial derivatives exist and are continuous on U. Equivalently, f is differentiable at every point with a linear approximation given by a continuous derivative map Df: U → L(R^n, R^m).

Examples include polynomials, exponential and trigonometric functions, and most standard smooth mappings. A function can be differentiable with a non-continuous derivative (and thus not be C^1); for instance, a classic example is a differentiable function whose derivative oscillates without limit.

C^1 functions enjoy several important properties: they are closed under addition, scalar multiplication, and composition; the chain rule applies to compositions of C^1 maps; and the inverse function theorem can guarantee local diffeomorphisms when the Jacobian determinant is nonzero. On compact subsets of their domain, C^1 functions are locally Lipschitz because the derivative is continuous and hence bounded there. In analysis and differential geometry, C^1 represents a basic level of regularity between continuity and higher smoothness (C^k, k ≥ 2).