Question

Show that the function $straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 3 close vertical bar comma space straight x element of bold R bold comma$ is continuous but not differentiable at x=3.

Solution

straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 3 close vertical bar space equals space open vertical bar table row cell 3 minus straight x comma space space space straight x less than 3 end cell row cell straight x minus 3 comma space straight x greater or equal than 3 end cell end table close vertical bar

Let c be a real number.
Case I: c<3 Then f(c) = 3-c.

limit as straight x rightwards arrow straight c of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow straight c of left parenthesis 3 minus straight x right parenthesis space equals space 3 minus straight c. Since comma space limit as straight x rightwards arrow straight c of straight f left parenthesis straight x right parenthesis space equals space straight f left parenthesis straight c right parenthesis comma space straight f space is space continous space at space all space negatives space real space numbers.

CaseII: c = 3. Then f(c) = 3 - 3 = 0

limit as straight x rightwards arrow straight c of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow straight c of left parenthesis straight x minus 3 right parenthesis space equals space 3 minus 3 space equals space 0

Since

limit as straight x rightwards arrow straight c of straight f left parenthesis straight x right parenthesis space equals space straight f left parenthesis 3 right parenthesis comma space

f is continuous at x = 3.
Case III: C>3. Then f(c) = c - 3

limit as straight x rightwards arrow straight c of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow straight c of left parenthesis straight x minus 3 right parenthesis space equals space straight c minus 3.

Since,

limit as straight x rightwards arrow straight c of left parenthesis straight x minus 3 right parenthesis space equals space straight c minus 3.

Therefore, f is a continuous function.
Now, we need to show that

straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 3 close vertical bar comma space straight x space element of space bold R bold space is space not space differentiable space at space straight x space equals space 3.

Consider the left hand limit of f at x = 3

limit as straight h rightwards arrow 0 to the power of minus of fraction numerator straight f left parenthesis 3 plus straight h right parenthesis minus straight f left parenthesis 3 right parenthesis over denominator straight h end fraction space equals space limit as straight h rightwards arrow 0 to the power of minus of fraction numerator open vertical bar 3 plus straight h minus 3 close vertical bar minus open vertical bar 3 minus 3 close vertical bar over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of minus of fraction numerator open vertical bar straight h close vertical bar minus 0 over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of minus of fraction numerator negative straight h over denominator straight h end fraction equals 1 left parenthesis straight h less than 0 space rightwards double arrow space open vertical bar straight h close vertical bar space equals space minus straight h right parenthesis

Consider the right hand limit of f at x = 3

limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight f left parenthesis 3 plus straight h right parenthesis minus straight f left parenthesis 3 right parenthesis over denominator straight h end fraction limit as straight h rightwards arrow 0 to the power of plus of fraction numerator open vertical bar 3 plus straight h minus 3 close vertical bar minus open vertical bar 3 minus 3 close vertical bar over denominator straight h end fraction space equals limit as straight h rightwards arrow 0 to the power of plus of fraction numerator open vertical bar straight h close vertical bar minus 0 over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of plus of straight h over straight h equals 1

left parenthesis straight h greater than 0 space rightwards double arrow space open vertical bar straight h close vertical bar space equals space straight h right parenthesis

Since the left and right hand limits are not equal, f is not differentiable at x = 3.

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