Please use JavaScript 5.5.4 version of MathGraph32 (ou a more recent one).

We want to graph an approximation of the blancmange function, continuous everywhere but differentiable nowhere (and periodical of period 1).

This function is defined by : $blanc(x)=\sum_{n=0}^{+\infty}\frac{1}{2^k}\left| 2^kx-\left[ 2^kx+\frac{1}{2} \right] \right|$ where [x] stands for integrer part of *x*.

Let us start with a new figurewmith an orthonormal frame with icon . Choose *Figure with frame* and ask for a frame with dots and without scaling of axes.

With capture icon ,drag point O toward the left bottom of the figure and drag point I away from O (just like the figure at the bottom of this article).

We will now create a cursor with integer values values of which will be named *k* with icon .

Click on the top left of the figure (this will be the left edge of the cursor) and fill in the dialog box as shown below (#I is for italic and #N for a return to a normal font).

You can use a bigger font for the display of the cursor value with tool (modification of graphical object).

Let us now create a function of two variables *g*. Expand the toolbar for cursors and calculations, click on icon and choose * Real function of two variables*. The formula for the function is `1/2^n*abs(2^n*x-int(2^n*x+1/2))`

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We go on with the creation of a function *fk* defined by $f_k(x)=\frac{1}{2^k}\left| 2^kx-\left[ 2^kx+\frac{1}{2} \right] \right|$.

In the color palette, activate the purple color then use icon (expand the horizontal toolbar with the cursor icon).

Fill in the dialog box as shown below :

The graph of function *fk* appears. If you capture the cursor with tool you see this curve change in real time.

Now we want to create function *f* (approximation of function *blanc* wich will be the sum of functions *fk* for *k* ranging from 0 to 10).

In the color palette, activate the blue color.

Use icon . Fill in the dialog box as shown below :

Here we ask for 2000 points because this curve is very complicated. Note that if you click oh the **predefined function** button you can see the syntax of the *sum* function.

To show how the approximation is fit, we will create a last function *h* and graph it.

In the color palette activate the green color.

Use icon . Fill in the dialog box as below :

Now move the cursor to change the value of *k* (icĂ´ne ).

To finish we can create three LaTeX displays in the same color as the curves with tool .

Here is the code for these LaTeX displays :

`f_k(x)=\frac{1}{2^k}\left| 2^kx-\left[ 2^kx+\frac{1}{2} \right] \right|`

`f(x)=\sum_{n=0}^{10}f_n(x)`

`g(x)=\sum_{n=0}^{k}f_n(x)`

And below the figure animated with MathGraph32 JavaScript library.