# 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. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.

More generally, for *x*_{0} as an interior point in the domain of a function *f*, then *f* is said to be *differentiable at x*_{0} if and only if the derivative *f* ′(*x*_{0}) exists. In other words, the graph of *f* has a non-vertical tangent line at the point (*x*_{0}, *f*(*x*_{0})). The function *f* is also called *locally linear* at *x*_{0} as it is well approximated by a linear function near this point.