Analysis.........the study of limits................bounds of what the value of the variables can be..........who knows....................



With more work, you can show that if –1.5 < a₀ < 1.5, then the iterates approach –0.5, like they did when a₀ was 1.0. The remaining two initial values, namely 1.5 and –1.5, have all their iterates equal to 1.5. So the set of real numbers is partitioned into two parts, the closed interval [–1.5,1.5] of initial values whose iterates remain bounded, and the rest of the real numbers whose iterates do not remain bounded but approach infinity. This is shown graphically below. The real line is partitioned into the two sets with the interval [–1.5,1.5] drawn in light yellow, the rest in pink. We'll call the set of initial values whose iterates remain bounded the filled-in Julia set, or simply the Julia set for short. Thus, the light yellow set is the real Julia set for the function f(x) = x² – 0.75.