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Math · Calculus
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Before you attempt the rest of the problems, you may find it useful to first study the standard example of a function with a discontinuous derivative. You may have seen this function in class already, but here is a refresher: https://tinyurl.com/mat137ps5 2. Consider the following FALSE theorem and BAD proof False theorem Let h be a function defined on an open interval I. Assume h is differentiable on I. Then h is continuous on I Bad proof h(x) - h(a) Let a E 1. By definition, h,(a) = lim Since h is continuous, the limit of the numerator is 0. The limit of the denominator is also 0. Since h is differentiable, I can apply LHôpitals Rule. h,(a) = lima- -a h)-ha) u lim x→a I have proven that h,(a) = lim h,(x). By definition, h, is continuous. ロ (a) Explain the error in the proof (b) Fix the theorem. (In other words, modify the statement of the theorem a little bit, either changing the hypotheses or the conclusion, so that it is true. There may be more than one way to do it.) You do not need to write the proof

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