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I readed the sample solution for the bonus question, which uses the limits approach to prove the continuity of the F(x)*G(x). I'm just wondering since i used the approach of derivative to prove it. Based on the Theorem 3.6, page 146, "If a function has a derivative at x=a, then the function is continuous at x=a."

I assume the derivatives of F(x) and G(x) at point x=a are F'(a)=M and G'(a)=N , and value of F(a) and G(a) ,respectively. Then i differenate the function F(x)*G(x), which will get F'(a)*G(a)+F(a)*G'(a), which indicating the continuty of the function F(x)*G(x). Thanks for your time.

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Under the conditions in that problem you may not assume that the derivatives of F(x) and G(x) exist at x=a. It is true that when a function is differentiable at x=a then it is continuous there, but the converse is not true. So, differentiability of F(x) and G(x) at x=a can not be deduced from continuity of these two at x=a (and the latter is the only thing that is given for F and G).

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thx, professor Sasho. i will apply the right approach to the right question for the rest exams, thank you for time. By the way, just one more thing, will there also be bonus question in the final exam? Thanks again.

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i'm looking forward to it.

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>will there also be bonus question in the final exam?

I do not know yet.