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) is defined. 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 Differentiable Function: Differentiability and Continuity, Differentiability Classes, Differentiability in Higher Dimensions, Differentiability in Complex Analysis, Differentiable Functions On Manifolds
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—Walter Benjamin (1892–1940)