Question

Show that the function f given by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x cubed plus 3 comma space if space straight x not equal to 0 end cell row cell 1 comma space space space space space space space space space if space straight x not equal to 0 end cell end table close$
is not continuous at x = 0.

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x cubed plus 3 comma space straight x not equal to 0 end cell row cell 1 space space space space space space space space comma space straight x equals 0 end cell end table close space space Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 below left parenthesis straight x cubed plus 3 right parenthesis equals left parenthesis 0 right parenthesis cubed plus 3 equals 0 plus 3 equals 3

Also f is defined at x = 0

and space straight f left parenthesis 0 right parenthesis equals 1 therefore space Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals 1

∴ f is discontinuous at x = 0.

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