JSXGraph
JSXGraph share

Share

Discontinuous derivative
QR code 
<iframe 
    src="http://jsxgraph.uni-bayreuth.de/share/iframe/discontinuous-derivative" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Discontinuous derivative" 
    allowfullscreen
></iframe>
This code has to 
<div id="board-0-wrapper" class="jxgbox-wrapper " style="width: 100%; ">
   <div id="board-0" class="jxgbox" style="aspect-ratio: 1 / 1; width: 100%;" data-ar="1 / 1"></div>
</div>
<div id="board-1-wrapper" class="jxgbox-wrapper " style="width: 100%; ">
   <div id="board-1" class="jxgbox" style="aspect-ratio: 1 / 1; width: 100%;" data-ar="1 / 1"></div>
</div>
 
<script type = "text/javascript"> 
    /*
    This example is licensed under a 
    Creative Commons Attribution 4.0 International License.
    https://creativecommons.org/licenses/by/4.0/
    
    Please note you have to mention 
    The Center of Mobile Learning with Digital Technology
    in the credits.
    */
    
    const BOARDID0 = 'board-0';
    const BOARDID1 = 'board-1';
    const BOARDID = BOARDID0;
 
    (function() {
        const board = JXG.JSXGraph.initBoard(BOARDID0, {axis:true, boundingbox:[-1/2, 1/2, 1/2, -1/2]});
    
        var g = board.create('functiongraph', ["2*x*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
        var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
    })();
    
    (function() {
        const board = JXG.JSXGraph.initBoard(BOARDID1, {axis:true, boundingbox:[-1/2, 0.08, 1.5, -0.02]});
    
        var g_der = board.create('functiongraph', [
            "(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], 
            {strokeColor: 'red'});
        var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"],  
            {strokeWidth:2});
    })();
    
 </script> 
/*
This example is licensed under a 
Creative Commons Attribution 4.0 International License.
https://creativecommons.org/licenses/by/4.0/
 
Please note you have to mention 
The Center of Mobile Learning with Digital Technology
in the credits.
*/
 
const BOARDID0 = 'your_div_id_0'; // Insert your 1st board id here!
const BOARDID1 = 'your_div_id_1'; // Insert your 2nd board id here!
 
(function() {
    const board = JXG.JSXGraph.initBoard(BOARDID0, {axis:true, boundingbox:[-1/2, 1/2, 1/2, -1/2]});
 
    var g = board.create('functiongraph', ["2*x*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
    var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
})();
 
(function() {
    const board = JXG.JSXGraph.initBoard(BOARDID1, {axis:true, boundingbox:[-1/2, 0.08, 1.5, -0.02]});
 
    var g_der = board.create('functiongraph', [
        "(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], 
        {strokeColor: 'red'});
    var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"],  
        {strokeWidth:2});
})();
<jsxgraph width="100%, 100%" aspect-ratio="1 / 1, 1 / 1" numberOfBoards="2" title="Discontinuous derivative" description="This construction was copied from JSXGraph examples database: BTW HERE SHOULD BE A GENERATED LINKuseGlobalJS="false">
   /*
   This example is licensed under a 
   Creative Commons Attribution 4.0 International License.
   https://creativecommons.org/licenses/by/4.0/
   
   Please note you have to mention 
   The Center of Mobile Learning with Digital Technology
   in the credits.
   */
   
   (function() {
       const board = JXG.JSXGraph.initBoard(BOARDID0, {axis:true, boundingbox:[-1/2, 1/2, 1/2, -1/2]});
   
       var g = board.create('functiongraph', ["2*x*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
       var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
   })();
   
   (function() {
       const board = JXG.JSXGraph.initBoard(BOARDID1, {axis:true, boundingbox:[-1/2, 0.08, 1.5, -0.02]});
   
       var g_der = board.create('functiongraph', [
           "(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], 
           {strokeColor: 'red'});
       var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"],  
           {strokeWidth:2});
   })();
   
</jsxgraph>

Examples of continuous, differentiable functions with discontinuous derivative.

First example

Consider the function (blue curve)

f:R→R,x↦{x2sin(1/x),x≠00,x=0. ff is a continuous and differentiable function. The derivative of ff is the function (red curve)

f′:R→R,x↦{2xsin(1/x)−cos(1/x),x≠00,x=0. We observe that f′(0)=0f'(0) = 0, but limx→0f′(x)\lim_{x\to0}f'(x) does not exist.

Therefore, f′f' is an example of a derivative which is not continuous.

Second example

g:R→R,x↦{x2(1−x)2sin(1πx(1−x)),0<x<10,otherwise.

Here, the derivative is not continuous for x=0x=0 and x=1x=1.

0.10.20.30.4−0.1−0.2−0.3−0.40.10.20.30.4−0.1−0.2−0.3−0.4
0.20.40.60.811.21.4−0.2−0.40.010.020.030.040.050.060.07−0.01
// Define the ids of your boards in BOARDID0, BOARDID1,...
 
(function() {
    const board = JXG.JSXGraph.initBoard(BOARDID0, {axis:true, boundingbox:[-1/2, 1/2, 1/2, -1/2]});
 
    var g = board.create('functiongraph', ["2*x*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
    var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
})();
 
(function() {
    const board = JXG.JSXGraph.initBoard(BOARDID1, {axis:true, boundingbox:[-1/2, 0.08, 1.5, -0.02]});
 
    var g_der = board.create('functiongraph', [
        "(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], 
        {strokeColor: 'red'});
    var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"],  
        {strokeWidth:2});
})();

license

This example is licensed under a Creative Commons Attribution 4.0 International License.
Please note you have to mention The Center of Mobile Learning with Digital Technology in the credits.