Question

Let R be the set of real numbers and f : R → R be such that for all x and y in R in R

Answer 1

Answer:

f(x)=±2x+C0

Explanation:

If |f(x)−f(y)|=2|x−y| then f(x) is Lipschitz continuous. So the function f(x) is differentiable. Then following,

|f(x)−f(y)||x−y|=2 or
∣∣∣f(x)−f(y)x−y∣∣∣=2 now

limx→y∣∣∣f(x)−f(y)x−y∣∣∣=∣∣∣limx→yf(x)−f(y)x−y∣∣∣=|f'(y)|=2

so

f(x)=±2x+C0


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