How to "Repeat" A Function In an Interval Periodically? Given a function $f(x)$, how can you use sine (and not mod), to make $f(x)$ in an interval of $[a,b]$ to repeat every interval of $b-a$? In answers, use $f(x)=x$ as an example. I also do not expect it to be continuous.
How about
$$f\left(\frac{a+b}2+\frac {b-a}\pi \arctan \left  ( \tan \left ( -\frac \pi 2 +\pi\frac {x-a} {b-a} \right)\right)\right)$$
You want the period to be $b-a=T,$ so compose the function with a sinusoid with this period.