Question

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

Solution

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

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

space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight c below left parenthesis 2 straight x plus 3 right parenthesis equals 2 straight c plus 3 Also space space space space space straight f left parenthesis straight c right parenthesis equals 2 straight c plus 3

∴ f is continuous at all points .x < 2.

Case II : c > 2
$space space space space space space space space space stack space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight c below left parenthesis 2 straight x minus 3 right parenthesis equals 2 straight c minus 3 Also space space space space space space space space space straight f left parenthesis straight c right parenthesis equals 2 straight c minus 3 therefore space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f is continuous at all points x > 2.

Case III : x = 2
$space space space Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of minus below left parenthesis 2 straight x plus 3 right parenthesis equals 4 plus 3 equals 7 space space space Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 2 to the power of plus below left parenthesis 2 straight x minus 3 right parenthesis equals 4 minus 3 equals 1 therefore Lt with straight x rightwards arrow 2 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis$
∴ f is not continuous at x = 2
∴ x = 2 is the only point of discontinuity of f.

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