analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers. So, a function is said to be differentiable at when Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function , that is complex-differentiable at a point is automatically differentiable at that point, when viewed as a function . This is because the complex-differentiability implies that However, a function can be differentiable as a multi-variable function, while not being complex-differentiable. For example, is differentiable at every point, viewed as the 2-variable real function , but it is not complex-differentiable at any point. Any function that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Such a function is necessarily infinitely differentiable, and in fact analytic. Differentiable functions on manifolds If M is a differentiable manifold, a real or complex-valued function f on M is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate chart defined around p. If M and N are differentiable manifolds, a function f: M → N is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate charts defined around p and f(p). See also Generalizations of the derivative Semi-differentiability Differentiable programming References Differential calculus Multivariable calculus Smooth functions