Discontinuous derivative: Difference between revisions

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Consider the function
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
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.
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 09:02, 13 February 2019

Consider the function (blue curve)

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

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

Therefore, [math]\displaystyle{ 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});