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Continuity And Differentiability

Question

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Prove that the function f given by f(x) = |x - 1 |, x ∈ R is not differentiable at x = 1.

Solution

straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 1 close vertical bar straight L. straight H. straight D equals Lt with straight x rightwards arrow 1 to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 1 right parenthesis over denominator straight x minus 1 end fraction equals Lt with straight x rightwards arrow 1 to the power of minus below fraction numerator open vertical bar straight x minus 1 close vertical bar minus open vertical bar 1 minus 1 close vertical bar over denominator straight x minus 1 end fraction space space space space space space space space equals Lt with straight x rightwards arrow 1 to the power of minus below fraction numerator open vertical bar straight x minus 1 close vertical bar over denominator straight x minus 1 end fraction space space space space space space space left square bracket Put space straight x equals 1 minus straight h comma space straight h greater than 0 comma space so space that space straight h rightwards arrow 0 comma space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 1 to the power of minus right square bracket space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 1 minus straight h minus 1 close vertical bar over denominator 1 minus straight h minus 1 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar negative straight h close vertical bar over denominator negative straight h end fraction space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator negative straight h end fraction equals Lt with straight h rightwards arrow 0 below left parenthesis negative 1 right parenthesis equals negative 1 straight R. straight H. straight D equals Lt with straight x rightwards arrow 1 to the power of plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 1 right parenthesis over denominator straight x minus 1 end fraction equals Lt with straight x rightwards arrow 1 to the power of plus below fraction numerator open vertical bar straight x minus 1 close vertical bar minus open vertical bar 1 minus 1 close vertical bar over denominator straight x minus 1 end fraction space space space space space space space space equals Lt with straight x rightwards arrow 1 to the power of plus below fraction numerator open vertical bar straight x minus 1 close vertical bar over denominator straight x minus 1 end fraction space space space space space space space left square bracket Put space straight x equals 1 plus straight h comma space straight h greater than 0 comma space so space that space straight h rightwards arrow 0 comma space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 1 to the power of plus right square bracket space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 1 plus straight h minus 1 close vertical bar over denominator 1 plus straight h minus 1 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight h close vertical bar over denominator straight h end fraction space space space space space space space space equals Lt with straight h rightwards arrow 0 below straight h over straight h equals Lt with straight h rightwards arrow 0 below left parenthesis 1 right parenthesis equals 1 therefore straight L. straight H. straight D not equal to straight R. straight H. straight D therefore space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 1 close vertical bar space is space not space differentiable space at space straight x equals 1.

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