Difference between revisions of "Discontinuous derivative"

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Consider the function
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Consider the function (blue curve)
 
:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto  
 
:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto  
 
\begin{cases}
 
\begin{cases}
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</math>
 
</math>
 
<math>f</math> is a continous and differentiable.
 
<math>f</math> is a continous and differentiable.
The derivative of <math>f</math> is the function
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The derivative of <math>f</math> is the function (red curve)
 
:<math>
 
:<math>
 
f': \mathbb{R} \to \mathbb{R}, x \mapsto
 
f': \mathbb{R} \to \mathbb{R}, x \mapsto
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\end{cases}\,.
 
\end{cases}\,.
 
</math>
 
</math>
We observe that <math>f'(0) = 0</math> but <math>\lim_{x\to0}f'(x)</math> does not exist.
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We observe that <math>f'(0) = 0</math>, but <math>\lim_{x\to0}f'(x)</math> does not exist.
  
 
Therefore, <math>f'</math> is an example of a derivative which is not continuous.
 
Therefore, <math>f'</math> is an example of a derivative which is not continuous.

Revision as of 11:02, 13 February 2019

Consider the function (blue curve)

[math] f: \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} x^2\sin(1/x),& x\neq 0\\ 0,& x=0 \end{cases}\,. [/math]

[math]f[/math] is a continous and differentiable. The derivative of [math]f[/math] is the function (red curve)

[math] f': \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} 2\sin(1/x) - \cos(1/x), &x \neq 0\\ 0,& x=0 \end{cases}\,. [/math]

We observe that [math]f'(0) = 0[/math], but [math]\lim_{x\to0}f'(x)[/math] does not exist.

Therefore, [math]f'[/math] is an example of a derivative which is not continuous.

The underlying JavaScript code

var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});

var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});