<P> 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 . </P> <P> More generally, if x is a point in the domain of a function f, then f is said to be differentiable at x if the derivative f ′ (x) exists . This means that the graph of f has a non-vertical tangent line at the point (x, f (x)). The function f may also be called locally linear at x, as it can be well approximated by a linear function near this point . </P> <P> If f is differentiable at a point x, then f must also be continuous at x . 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 . </P> <P> Most functions that occur in practice have derivatives at all points or at almost every point . However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions . Informally, this means that differentiable functions are very atypical among continuous functions . The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function . </P>

If a function is continuous then it is differentiable at that point