Suppose f is continuously differentiable on an interval (a, 6). Prove that on any closed subinterval [c, d] the function is uniformly dif- ferentiable in the sense that given any 1/n there exists 1/m (inde- pendent of ro) such that |f(x)-f(xo)-f'(x0)(x-xo)|I < lx– x0|/n whenever r - xo] < 1/m. (Hint: use the mean value theorem and the uniform continuity of f' on [c, d].)

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