Recall the Takagi function $\tau$ that we discussed in class - it was defined as follows: We first defined $\varphi(x)=|x|$ for $-1/2\leq x \leq 1/2$ and extended $\varphi$ to periodic. This creates a basic sawtooth wave. We then defined $\tau$ in terms of $\varphi$ by

$$\tau(x) = \sum_{n=0}^\infty \frac{1}{2^n}\varphi(2^n x).$$

We can plot this thing, along with the line $y=x$ quite easily:

```
phi[x_] := With[{y = Mod[x, 1]},
If[y <= 1/2, y, 1 - y]];
tau[x_Real] := Sum[phi[2^k*x]/2^k, {k, 0, 53}];
Plot[{tau[x], x}, {x, 0, 1}, PlotRange -> {0, 0.75}]
```

Note that it appears that the graphs intersect at the cusps where $x=1/2$, $x=3/4$ and more. Those are fixed points of $\tau$.