Question

Find all points of discontinuity of f, where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x cubed minus 3 comma space if space straight x less or equal than 2 end cell row cell straight x squared plus 1 comma space if space straight x greater than 2 end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x cubed minus 3 comma space if space straight x less or equal than 2 end cell row cell straight x squared plus 1 comma space if space straight x greater than 2 end cell end table close

Function f is defined at all points of the real line.
Let c be any real number.
Three cases arise ;
Case I ; c < 2

space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis straight x cubed minus 3 right parenthesis equals straight c cubed minus 3 Also space space space straight f left parenthesis straight c right parenthesis equals straight c cubed minus 3 therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at all points x < 2.
Case II : c > 2

space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis straight x squared plus 1 right parenthesis equals straight c squared plus 1 Also space space space straight f left parenthesis straight c right parenthesis equals straight c squared plus 1 therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at all points x > 2
CaseIII ; c = 2

Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of minus below left parenthesis straight x cubed minus 3 right parenthesis equals 8 minus 3 equals 5 Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of plus below left parenthesis straight x squared plus 1 right parenthesis equals 4 plus 1 equals 5 Also space straight f left parenthesis 2 right parenthesis equals left parenthesis 2 right parenthesis cubed minus 3 equals 8 minus 3 equals 5 therefore space Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 2 right parenthesis

∴ f is continuous at x = 2.

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