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SOLVED:Let I be an interval. Prove that if f is differentiable on I and if the derivative f^' is bounded on I, then f satisfies a Lipschitz condition on I. (See Definition 5.4.4.)
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SOLVED:(a) Use the Mean-Value Theorem to show that if f is differentiable on an interval, and if |f^'(x)| ≤M for all values of x in the interval, then |f(x)-f(y)| ≤M|x-y| for all values of x and y in the interval. (b) Use the result in part (a) to show that |sinx-siny| ≤|x-y| for all real values of x and y .
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