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Continuity And Differentiability

Question

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Solution

(i) Let f(x) = [x], 0 < x < 3.

Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of minus below open square brackets straight x close square brackets 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 space space space space space equals Lt with straight h rightwards arrow 0 below open square brackets 1 minus straight h close square brackets equals Lt with straight h rightwards arrow 0 below left parenthesis 0 right parenthesis equals 0 Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below open square brackets straight x close square brackets 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 space space space space space space equals Lt with straight h rightwards arrow 0 below open square brackets 1 plus straight h close square brackets equals Lt with straight h rightwards arrow 0 below left parenthesis 1 right parenthesis equals 1 therefore space space Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis rightwards double arrow space straight f space is space not space continous space at space straight x equals 1 rightwards double arrow space straight f space is space not space differentiable space at space straight x equals 1.

(ii) Let f(x) = [x], 0 < x < 3.

Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of minus below open square brackets straight x close square brackets space space space space left square bracket Put space straight x equals 2 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 2 to the power of minus right square bracket space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below open square brackets 2 minus straight h close square brackets equals Lt with straight h rightwards arrow 0 below left parenthesis 1 right parenthesis equals 1 Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of plus below open square brackets straight x close square brackets space space space space left square bracket Put space straight x equals 2 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 2 to the power of plus right square bracket space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below open square brackets 2 plus straight h close square brackets equals Lt with straight h rightwards arrow 0 below left parenthesis 2 right parenthesis equals 2 therefore space Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis therefore space straight f space is space not space continous space at space straight x equals 2. rightwards double arrow space straight f space is space not space differentiable space at space straight x equals 2.

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