Question

Discuss the continuity of the functions

$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell space space space space space space straight x space space comma space space space straight x greater than 0 end cell row cell space space space space space 1 space space space comma space space space straight x equals 0 end cell row cell space minus straight x space space space comma space space space straight x less than 0 end cell end table close at space straight x equals 0$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell space space space space space space straight x space space comma space space space straight x greater than 0 end cell row cell space space space space space 1 space space space comma space space space straight x equals 0 end cell row cell space minus straight x space space space comma space space space straight x less 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 space Lt with straight x rightwards arrow 0 to the power of minus below left parenthesis negative 0 right parenthesis equals space Lt with straight h rightwards arrow 0 below left curly bracket negative left parenthesis 0 minus straight h right parenthesis 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 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 h 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 space equals space Lt with straight h rightwards arrow 0 below straight h equals 0 But space straight f left parenthesis 0 right parenthesis equals 1 therefore space space Lt with straight x rightwards arrow 0 below straight f left parenthesis 0 right parenthesis not equal to straight f left parenthesis 0 right parenthesis 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 rightwards double arrow straight f left parenthesis straight x right parenthesis space is space discontinous space at space straight x equals 0

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