Question

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

Solution

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

space space space space space Lt with space space space 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 straight f left parenthesis straight x squared plus 1 right parenthesis equals straight c squared plus 1 space space space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c squared plus 1 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 points x < 1.
Case II : c > 1

space space space space space Lt with space space space 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 straight f left parenthesis straight x plus 1 right parenthesis equals straight c plus 1 space space space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c plus 1 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 points x > 1.
Case III ; c = 1

space space space space space Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of minus below left parenthesis straight x squared plus 1 right parenthesis equals 1 plus 1 equals 2 space space space space space Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below left parenthesis straight x plus 1 right parenthesis equals 1 plus 1 equals 2 space space space space space space space space space space space space space space straight f left parenthesis 1 right parenthesis equals 1 plus 1 equals 2 therefore Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 1 right parenthesis

∴ f is continuous at x = 1
∴ f is continuous at all points of domain.

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