Difference between revisions of "Darboux function"

From specialfunctionswiki
Jump to: navigation, search
(Created page with "The Darboux function is defined by $$D(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin\left((k+1)!x\right)}{k!},$$ where $\sin$ denotes the sine function. =Properties= <...")
 
 
(6 intermediate revisions by the same user not shown)
Line 3: Line 3:
 
where $\sin$ denotes the [[sine]] function.
 
where $\sin$ denotes the [[sine]] function.
  
=Properties=
+
<div align="center">
<div class="toccolours mw-collapsible mw-collapsed">
+
<gallery>
<strong>Theorem:</strong> The Darboux function is [[continuous]].
+
File:Darbouxplot.png|Plot of $D(x)$ on $[0,5]$.
<div class="mw-collapsible-content">
+
</gallery>
<strong>Proof:</strong> █
 
</div>
 
 
</div>
 
</div>
  
<div class="toccolours mw-collapsible mw-collapsed">
+
=Properties=
<strong>Theorem:</strong> The Darboux function is [[nowhere differentiable]].
+
[[Darboux function is continuous]]<br />
<div class="mw-collapsible-content">
+
[[Darboux function is nowhere differentiable]]<br />
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=References=
 
=References=
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br />
+
* {{BookReference|Continuous Nowhere Differentiable Functions|2003|Johan Thim|prev=findme|next=Schwarz function}} $\S 3.5$, pg. 28
 +
 
 +
{{:Continuous nowhere differentiable functions footer}}
 +
 
 +
[[Category:SpecialFunction]]

Latest revision as of 18:02, 25 June 2017

The Darboux function is defined by $$D(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin\left((k+1)!x\right)}{k!},$$ where $\sin$ denotes the sine function.

Properties

Darboux function is continuous
Darboux function is nowhere differentiable

References

Continuous nowhere differentiable functions