Question

Which of the following functions are continuous at the indicated points ?
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space space space comma space if space straight x less than 0 end cell row cell straight x squared space space comma space if space straight x greater than 0 end cell end table close$

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Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space space space comma space if space straight x less than 0 end cell row cell straight x squared space space comma space if space straight x greater than 0 end cell end table close space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below straight x equals Lt with straight h rightwards arrow 0 below left parenthesis 0 minus straight h right parenthesis 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 0 minus straight h comma space straight h greater than 0 right square bracket space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below left parenthesis negative straight h right parenthesis equals 0 space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below straight x squared equals Lt with straight h rightwards arrow 0 below left parenthesis 0 plus straight h squared right parenthesis space space space space space space space space 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 right square bracket space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below straight h squared equals 0 Also space straight f left parenthesis 0 right parenthesis equals left parenthesis 0 right parenthesis squared equals 0 therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals left parenthesis 0 right parenthesis squared equals 0 therefore space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0.

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