Question

Which of the following functions are continuous at the indicated points ?
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space straight x less or equal than 0 end cell row cell 2 straight x plus 1 space space space comma space space space if space straight x greater or equal than 0 end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space straight x less or equal than 0 end cell row cell 2 straight x plus 1 space space space comma space space space if space straight x greater or equal 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 left parenthesis 2 straight x minus 1 right parenthesis equals Lt with straight h rightwards arrow 0 below left curly bracket 2 left parenthesis 0 minus straight h right parenthesis minus 1 right curly bracket 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 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 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 space space space space space space equals Lt with straight h rightwards arrow 0 below left parenthesis negative 2 straight h minus 1 right parenthesis equals 0 minus 1 equals negative 1 But space straight f left parenthesis 0 right parenthesis equals Value space of space left parenthesis 2 straight x plus 1 right parenthesis space at space straight x equals 0. equals 2.0 plus 1 equals 1 therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 0 right parenthesis

⇒ f(x) is discontinuous at x = 0.

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