Differentiability

In calculus, 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 interior point in its domain, be relatively smooth, and cannot contain any break, angle, or cusp.

More generally, if x₀ is an interior 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.