Differentiable function

A function is differentiable at a point when its change near that point is approximated by a linear map with an error negligible relative to the input change. Its derivative is that linear map.

Limit of a function: A value that function outputs approach as inputs approach a specified point. Differentiability is defined through a limit of the linear-approximation error.
Derivative: A quantity describing a function's instantaneous rate of change, or its local linear approximation. It is the linear map that defines differentiability at a point.
Taylor's theorem: A theorem approximating a sufficiently differentiable function by a polynomial with a controlled remainder. Differentiability of higher orders enables increasingly accurate local polynomial models.
Continuous nowhere-differentiable function: A function that is continuous at every point but differentiable at none. It demonstrates that continuity does not imply differentiability.
Linear map: A function between vector spaces that preserves addition and scalar multiplication. The derivative is the linear map that best describes local change.
Difference quotient: A ratio of a function's change to a corresponding change in its input. In one variable, its limit gives the derivative when that limit exists.
Newton's method: An iterative method for finding a function's root using tangent-line approximations. Its update step relies on the derivative to linearize the root-finding problem.
Weierstrass function: A continuous function constructed by Weierstrass that is nowhere differentiable. It is a canonical explicit example separating continuity from differentiability.
Absolute value function: The real function that assigns each number its nonnegative magnitude. Its corner at zero makes it continuous there but not differentiable.
Fréchet derivative: A bounded linear map that approximates a function locally with error negligible relative to input distance. This is the general linear-approximation definition for functions between normed spaces.

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