Let f: RR be a differentiable function then for any a, b E R, ab, the Lagrange's theorem says that f(b) f(a) = f'(c)(b-a), where is some number between a and b. Using this fact, prove that if f(x) is bounded, namely f'(c) < M, for some M> 0 and for all then f is a Lipschitz function. 2. Using the above result, prove that sin r. cos r. arctan z. ln(12²) is Lipschitz function.

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A: We have to use division algorithm property.

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A: Yes true.

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A: A homeomorphism is a continuous bijection with continuous inverse.

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