Suppose I have some strictly increasing function $f:[0,b]\to[0,b]$ with $0<b<1$, $f(x)<x$ and $f'(b)=\frac{1-f(b)}{1-b}$ (i.e. tangent to the secant line to $(1,1)$). Now imagine a 'tunnel' of width $\epsilon$ around it. I want to define a zig zag function of the following shape:
Starting at $(f(b),b)$, go down always on a secant either to $(1,1)$ or $(0,0)$, changing slopes every time you hit the boundary.
A picture can explain it better than words:
Is there a parsimonious way to formally define this line?
Edit: For the zig-zag function to be well-defined, assume also that the function $f$ needs to have increasing tangent slope to $(1,1)$ and to $(0,0)$, i.e. $$\frac{f(x)}{x}\text{ and }\frac{1-f(x)}{1-x}$$ are both increasing in $x$.