Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

More generally, if x₀ is a point in the domain of a function f, then f is said to be differentiable at x₀ if the derivative f ′(x₀) exists. This means that the graph of f has a non-vertical tangent line at the point (x₀, f(x₀)). The function f may also be called locally linear at x₀, as it can be well approximated by a linear function near this point.

Differentiability and continuity

If f is differentiable at a point x₀, then f must also be continuous at x₀. In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.