Question

Find all points of discontinuity of f, where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell open vertical bar straight x close vertical bar plus 3 comma space if space straight x less or equal than negative 3 end cell row cell negative 2 straight x comma if space minus 3 less than straight x greater than 3 end cell row cell 6 straight x plus 2 space space comma space if space straight x greater or equal than 3 end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell open vertical bar straight x close vertical bar plus 3 comma space if space straight x less or equal than negative 3 end cell row cell negative 2 straight x comma if space minus 3 less than straight x greater than 3 end cell row cell 6 straight x plus 2 space space comma space if space straight x greater or equal than 3 end cell end table close

Function f is defined for all points of the real line.
Let c be any real number.
Five cases arise :
Case I: c < –3

$space space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis open vertical bar straight x close vertical bar plus 3 right parenthesis equals open vertical bar straight c close vertical bar plus 3 Also space space space space space space space straight f left parenthesis straight c right parenthesis equals open vertical bar straight c close vertical bar plus 3 therefore space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f is continuous at all point x < – 3
Case II: c = –3

$Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of minus below left parenthesis open vertical bar straight x close vertical bar plus 3 right parenthesis equals open vertical bar negative 3 close vertical bar plus 3 equals 3 plus 3 equals 6 Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below left parenthesis negative 2 straight x right parenthesis equals 6 Also space straight f left parenthesis straight c right parenthesis equals straight f left parenthesis negative 3 right parenthesis equals open vertical bar negative 3 close vertical bar plus 3 equals 3 plus 3 plus equals 6 therefore space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis equals 6$
∴ f is continuous at x = – 3
Case III: – 3 < c < 3
f(x) = – 2 x is a continuous function as it is a polynomial.
Case IV : c = 3

space space space space space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of minus below left parenthesis negative 2 straight x right parenthesis equals negative 6 space space space space space Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below left parenthesis 6 straight x plus 2 right parenthesis equals 18 plus 2 equals 20 therefore space space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis

∴ f is discontinuous at x = 3.
Case V ; c > 3

$space space space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis 6 straight x plus 2 right parenthesis equals 6 straight c plus 2 Also space space space space space space space straight f left parenthesis straight c right parenthesis equals 6 straight c plus 2 therefore space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f is continuous at all points x > 3.

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