linguistic:voice_change
Relation to differentiability and integrability

Every differentiable function f: (a, b) → R {\displaystyle f:(a,b)\to \mathbb {R} } is continuous, as can be shown. The converse does not hold: for example, the absolute value function

is everywhere continuous. However, it is not differentiable at x = 0 {\displaystyle x=0} (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.

The derivative f′ (x) of a differentiable function f (x) need not be continuous. If f′ (x) is continuous, f (x) is said to be continuously differentiable. The set of such functions is denoted C 1 ((a, b)). {\displaystyle C^{1}((a,b)).} More generally, the set of functions f: Ω → R {\displaystyle f:\Omega \to \mathbb {R} } (from an open interval (or open subset of R {\displaystyle \mathbb {R} }) Ω {\displaystyle \Omega } to the reals) such that f is n {\displaystyle n} times differentiable and such that the n {\displaystyle n} -th derivative of f is continuous is denoted C n (Ω). {\displaystyle C^{n}(\Omega).} See differentiability class. In the field of computer graphics, properties related (but not identical) to C 0, C 1, C 2 {\displaystyle C^{0},C^{1},C^{2}} are sometimes called G 0 {\displaystyle G^{0}} (continuity of position), G 1 {\displaystyle G^{1}} (continuity of tangency), and G 2 {\displaystyle G^{2}} (continuity of curvature); see Smoothness of curves and surfaces.

Every continuous function f: → R {\displaystyle f:\to \mathbb {R} } is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable but discontinuous) sign function shows.