<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> <P> A function f is said to be continuously differentiable if the derivative f ′ (x) exists and is itself a continuous function . Though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity . For example, the function </P> <Dl> <Dd> f (x) = (x 2 sin ⁡ (1 / x) if x ≠ 0 0 if x = 0 (\ displaystyle f (x) \; = \; (\ begin (cases) x ^ (2) \ sin (1 / x) & (\ text (if)) x \ neq 0 \ \ 0& (\ text (if)) x = 0 \ end (cases))) </Dd> </Dl> <Dd> f (x) = (x 2 sin ⁡ (1 / x) if x ≠ 0 0 if x = 0 (\ displaystyle f (x) \; = \; (\ begin (cases) x ^ (2) \ sin (1 / x) & (\ text (if)) x \ neq 0 \ \ 0& (\ text (if)) x = 0 \ end (cases))) </Dd>

When is a function said to be differentiable