Defn: A function f:A→R for A⊆R is called Lipschitz if there is a bound M > 0 so that ((|f(x)−f(y)|) / (|x−y|)) ≤M for all x not equal to y with x,y∈A.

Show that f(x) =x+ sin(x) is Lipschitz on R.

(Hint:sin(x)−sin(y) = 2 cos((x+y)/2) sin((x−y)/2) and think about how|sinθ|and |θ|compare.)

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