Editing Continuous function (section)

====Relation to differentiability and integrability====
Every [[differentiable function]]
<math display="block">f : (a, b) \to \R</math>
is continuous, as can be shown. The [[Theorem#Converse|converse]]  does not hold: for example, the [[absolute value]] function
:<math>f(x)=|x| = \begin{cases}
  \;\;\ x & \text{ if }x \geq 0\\
  -x & \text{ if }x < 0
\end{cases}</math>
is everywhere continuous. However, it is not differentiable at <math>x = 0</math> (but is so everywhere else). [[Weierstrass function|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 <math>C^1((a, b)).</math> More generally, the set of functions
<math display="block">f : \Omega \to \R</math>
(from an open interval (or [[open subset]] of <math>\R</math>) <math>\Omega</math> to the reals) such that ''f'' is <math>n</math> times differentiable and such that the <math>n</math>-th derivative of ''f'' is continuous is denoted <math>C^n(\Omega).</math> See [[differentiability class]]. In the field of computer graphics, properties related (but not identical) to <math>C^0, C^1, C^2</math> are sometimes called <math>G^0</math> (continuity of position), <math>G^1</math> (continuity of tangency), and <math>G^2</math> (continuity of curvature); see [[Smoothness#Smoothness of curves and surfaces|Smoothness of curves and surfaces]].

Every continuous function
<math display="block">f : [a, b] \to \R</math>
is [[integrable function|integrable]] (for example in the sense of the [[Riemann integral]]). The converse does not hold, as the (integrable but discontinuous) [[sign function]] shows.