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Continuity And Differentiability

Question

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Let a function f r be continuous at a. Show that the function g (x) = (x - a)f(x) is differentiable at x = a.

Solution

Since space straight f space is space continous space at space straight a therefore space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis Now space Lt with straight x rightwards arrow straight a below fraction numerator straight g left parenthesis straight x right parenthesis minus straight g left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction equals Lt with straight x rightwards arrow straight a below fraction numerator left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight x right parenthesis minus left parenthesis straight a minus straight a right parenthesis space straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space straight g left parenthesis straight x right parenthesis minus left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight x right parenthesis right square bracket space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow straight a below fraction numerator left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis therefore space straight g left parenthesis straight x right parenthesis space is space differentiable space at space straight x equals straight a.

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