(3) Let f: R → R be a differentiable function and that ƒ has a bounded derivative. (That means there exists B > 0 such t

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3 Let F R R Be A Differentiable Function And That F Has A Bounded Derivative That Means There Exists B 0 Such T 1 (55.42 KiB) Viewed 16 times

(3) Let f: R → R be a differentiable function and that ƒ has a bounded derivative. (That means there exists B > 0 such that |ƒ'(x)| ≤ B for all x € R.) (a) Show that f is uniformly continuous on R. (b) If B < 1, then there exists a unique fixed point of f. (Hint: consider a sequence xo, f(xo), f(f(xo)),... defined by repeatedly applying f to an arbitrary xo € R.)