derivables

Derivables are functions that possess a derivative at every point of their domain. In real analysis, a real-valued function f defined on an interval I is derivable at a point a if the limit of the difference quotient

(f(a+h) − f(a)) / h as h → 0 exists. When this limit exists for every a in I, f is said to be derivable (or differentiable) on I, and the derivative is the function a ↦ f′(a). In several variables, the notion extends: a function f: U ⊆ R^n → R is differentiable at a point if there exists a linear map that best approximates f near that point; the derivative is represented by the Jacobian matrix or gradient.

Typical examples include polynomials, exponential functions, and logarithms, which are derivable on their respective domains. The absolute value function |x| is not derivable at x = 0, where its left and right derivatives differ. The function f(x) = x^2 is derivable everywhere on the real line, with f′(x) = 2x.

Key properties include that differentiability implies continuity. The derivative itself need not be continuous; a function can be differentiable at every point in an interval while its derivative is not continuous there (an example is f(x) = x^2 sin(1/x) for x ≠ 0 with f(0) = 0).

Higher-order differentiability leads to smoothness classifications: if f′ exists and is differentiable, f is C^1; if higher derivatives exist and are continuous, f is C^k; if all derivatives exist and converge to their Taylor series, the function is analytic. Derivables thus form a foundational class in calculus, linking instantaneous rate of change to approximation by linear maps.