Editing Continuous function (section)

===Properties===

====A useful lemma====
Let <math>f(x)</math> be a function that is continuous at a point <math>x_0,</math> and <math>y_0</math> be a value such <math>f\left(x_0\right)\neq y_0.</math> Then <math>f(x)\neq y_0</math> throughout some neighbourhood of <math>x_0.</math><ref>{{citation|last=Brown|first=James Ward|title=Complex Variables and Applications|year=2009|publisher=McGraw Hill|edition=8th|page=54|isbn=978-0-07-305194-9}}</ref>

''Proof:'' By the definition of continuity, take <math>\varepsilon =\frac{|y_0-f(x_0)|}{2}>0</math> , then there exists <math>\delta>0</math> such that 
<math display="block">\left|f(x)-f(x_0)\right| < \frac{\left|y_0 - f(x_0)\right|}{2} \quad \text{ whenever } \quad |x-x_0| < \delta</math>
Suppose there is a point in the neighbourhood <math>|x-x_0|<\delta</math> for which <math>f(x)=y_0;</math> then we have the contradiction
<math display="block">\left|f(x_0)-y_0\right| < \frac{\left|f(x_0) - y_0\right|}{2}.</math>

====Intermediate value theorem====
The [[intermediate value theorem]] is an [[existence theorem]], based on the real number property of [[Real number#Completeness|completeness]], and states:

:If the real-valued function ''f'' is continuous on the [[Interval (mathematics)|closed interval]] <math>[a, b],</math> and ''k'' is some number between <math>f(a)</math> and <math>f(b),</math> then there is some number <math>c \in [a, b],</math> such that <math>f(c) = k.</math>

For example, if a child grows from 1&nbsp;m to 1.5&nbsp;m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25&nbsp;m.

As a consequence, if ''f'' is continuous on <math>[a, b]</math> and <math>f(a)</math> and <math>f(b)</math> differ in [[Sign (mathematics)|sign]], then, at some point <math>c \in [a, b],</math> <math>f(c)</math> must equal [[0 (number)|zero]].

====Extreme value theorem====
The [[extreme value theorem]] states that if a function ''f'' is defined on a closed interval <math>[a, b]</math> (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists <math>c \in [a, b]</math> with <math>f(c) \geq f(x)</math> for all <math>x \in [a, b].</math> The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval <math>(a, b)</math> (or any set that is not both closed and bounded), as, for example, the continuous function <math>f(x) = \frac{1}{x},</math> defined on the open interval (0,1), does not attain a maximum, being unbounded above.

====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.

====Pointwise and uniform limits====
[[File:Uniform continuity animation.gif|A sequence of continuous functions <math>f_n(x)</math> whose (pointwise) limit function <math>f(x)</math> is discontinuous. The convergence is not uniform.|right|thumb]]
Given a [[sequence (mathematics)|sequence]]
<math display="block">f_1, f_2, \dotsc : I \to \R</math>
of functions such that the limit
<math display="block">f(x) := \lim_{n \to \infty} f_n(x)</math>
exists for all <math>x \in D,</math>, the resulting function <math>f(x)</math> is referred to as the [[Pointwise convergence|pointwise limit]] of the sequence of functions  <math>\left(f_n\right)_{n \in N}.</math> The pointwise limit function need not be continuous, even if all functions <math>f_n</math> are continuous, as the animation at the right shows. However, ''f'' is continuous if all functions <math>f_n</math> are continuous and the sequence [[Uniform convergence|converges uniformly]], by the [[uniform convergence theorem]]. This theorem can be used to show that the [[exponential function]]s, [[logarithm]]s, [[square root]] function, and [[trigonometric function]]s are continuous.