Differentiabiliteetti

Differentiabiliteetti is the mathematical property of a function having a derivative, meaning that the function can be locally approximated by a linear map. It is a central concept in calculus and analysis, describing how smoothly a function changes at a point or on a region.

In one variable, a function f: R -> R is differentiable at a if the limit of (f(a+h) − f(a)) / h as h → 0 exists, and this limit is denoted f′(a). In several variables, differentiability is usually formulated as Fréchet differentiability: a function f: U ⊆ R^n → R^m is differentiable at a ∈ U if there exists a linear map Df(a): R^n → R^m such that

f(a+h) − f(a) − Df(a)[h] = o(‖h‖) as h → 0. Df(a) is called the derivative or Jacobian, and for scalar-valued functions its components give the gradient.

Key properties include:

- Differentiability implies continuity, but not vice versa.

- The derivative provides a best linear approximation to the function near the point.

- Higher-order differentiability leads to smoothness: C^k or real-analytic functions have derivatives of all required orders.

Examples: f(x) = |x| is not differentiable at x = 0; f(x) = x^2 is differentiable for all x with f′(x) = 2x. In several variables, f(x,y) = x^2 + y^2 is differentiable everywhere with gradient ∇f(x,y) = (2x, 2y).

Differentiabiliteetti also interacts with broader theories: Lipschitz functions are differentiable almost everywhere (Rademacher’s theorem), and differentiability is a prerequisite for applying Taylor’s theorem and many optimization methods. The concept distinguishes from weaker notions like Gateaux differentiability, which may not guarantee a linear approximation.