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Continuity And Differentiability

Question

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Discuss the continuity of the function f given by f(x) = | x | at x = 0.

Solution

Here space space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x close vertical bar Lt with straight x rightwards arrow 0 to the power of minus below space space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below space open vertical bar straight x close vertical bar space space space space space space space left square bracket Put space straight x equals 0 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 0 to the power of minus right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below space open vertical bar 0 minus straight h close vertical bar equals Lt with straight h rightwards arrow 0 below open vertical bar negative straight h close vertical bar equals Lt with straight h rightwards arrow 0 below space straight h equals 0 Lt with straight x rightwards arrow 0 to the power of plus below space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below space open vertical bar straight x close vertical bar space space space space space space space left square bracket Put space straight x equals 0 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 0 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 space open vertical bar 0 plus straight h close vertical bar equals Lt with straight h rightwards arrow 0 below space space open vertical bar straight h close vertical bar equals Lt with straight h rightwards arrow 0 below space straight h equals 0 Also space straight f space is space defined space at space straight x equals 0 and space space space space straight f left parenthesis 0 right parenthesis equals open vertical bar 0 close vertical bar equals 0 therefore space space Lt space straight f left parenthesis straight x right parenthesis equals Lt space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis therefore space space space straight f space is space continous space at space straight x equals 0.

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