Question

Does there exist a function which is continuous everywhere but not differentiable at exactly two points ? Justify your answer.

Solution

Consider the function f given by
f(x) = |x - 1| + |x - 2|
This function is continuous everywhere
Differentiability at x = 1
$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 plus open vertical bar straight x minus 2 close vertical bar minus open vertical bar 1 minus 1 close vertical bar minus 1 minus 2 over denominator straight x minus 1 end fraction space space space 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 plus open vertical bar straight x minus 2 close vertical bar minus 1 over denominator straight x minus 1 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 1 minus straight h minus 1 close vertical bar plus open vertical bar 1 minus straight h minus 2 close vertical bar minus 1 over denominator 1 minus straight h minus 1 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 left square bracket Put space straight x equals 1 minus straight h comma space straight h greater than 0 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 space space space equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar negative straight h close vertical bar plus open vertical bar negative 1 minus straight h close vertical bar minus 1 over denominator negative straight h end fraction Lt with straight h rightwards arrow 0 below fraction numerator straight h plus 1 plus straight h minus 1 over denominator negative straight h end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator 2 space straight h over denominator negative straight h end fraction equals negative 2 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 plus open vertical bar straight x minus 2 close vertical bar minus 1 over denominator straight x minus 1 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 left square bracket Put space straight x equals 1 plus straight h comma space straight h greater than 0 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 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 plus open vertical bar 1 plus straight h minus 2 close vertical bar minus 1 over denominator 1 plus straight h minus 1 end fraction Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight h close vertical bar plus open vertical bar negative left parenthesis 1 minus straight h right parenthesis close vertical bar minus 1 over denominator straight h end fraction space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight h plus 1 plus straight h minus 1 over denominator straight h end fraction equals Lt with straight h rightwards arrow 0 below 0 over straight h equals 0 therefore space straight L. straight H. straight D not equal to straight R. straight H. straight D therefore space function space straight f space is space not space dervable space at space straight x equals 1.$
Similarly f is not derivable at x = 2. Also f is differentiable at any other point.
Hence the result.

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