Local Linearity
In calculus (a branch of mathematics), a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain. As a result, the graph of a differentiable function must be relatively smooth, and cannot contain any breaks, bends, cusps, or any points with a vertical tangent.
More generally, if x0 is a point in the domain of a function ƒ, then ƒ is said to be differentiable at x0 if the derivative ƒ′(x0) exists. This means that the graph of ƒ has a non-vertical tangent line at the point (x0, ƒ(x0)). The function ƒ may also be called locally linear at x0, as it can be well approximated by a linear function near this point.
Read more about Local Linearity: Differentiability and Continuity, Differentiability Classes, Differentiability in Higher Dimensions, Differentiability in Complex Analysis, Differentiable Functions On Manifolds
Famous quotes containing the word local:
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)