[math-fun] Describe f(a/b)=a/(b+1)?

Dumb questions I¹m not sure even how to ask: could someone help describe or visualize the following function f? For rational x = a/b (in lowest terms) let f(x) = a/(b+1) So for instance we can say things like x/2 <= f(x) < x. Next, to extend f to real x, we observe that in sequences of rationals x_n ³converging on² x the denominators (almost always) grow, so that f(x_n) ³approaches² x. So let¹s define f as the identity on irrational values. What else can we say about f? Is it continuous? What about its derivative? Fourier transform? It¹s sure ³bumpy², but is it ³fractally²?