# Difference between revisions of "Discontinuous derivative"

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A WASSERMANN (talk | contribs) |
A WASSERMANN (talk | contribs) |
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<jsxgraph width="500" height="500" box="jxgbox2"> | <jsxgraph width="500" height="500" box="jxgbox2"> | ||

− | var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0. | + | var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]}); |

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

− | var f = board.create('functiongraph', ["x^2*(1-x)^2*sin(1/(PI* x*(1-x)))"], {strokeWidth:2}); | + | var f = board.create('functiongraph', ["(x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2}); |

</jsxgraph> | </jsxgraph> | ||

## Revision as of 11:21, 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.

Here is another example:

- [math] g: \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} x^2(1-x)^2\sin(1/(\pi x(1-x)),& 0\ltx\lt 1\\ 0,& \mbox{otherwise} \end{cases}\,. [/math]

### The underlying JavaScript code

First example:

```
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});
```