Question

Show that the function of given by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x comma space space space space if space straight x greater or equal than 1 end cell row cell straight x squared comma space if space straight x less than 1 end cell end table close$

Solution

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

D_f = R. Let a be any real number. Three cases axis :
Case I: a > 1
In this case, all nearby points of a are also .> 1

Lt with straight x rightwards arrow straight a below equals Lt with straight x rightwards arrow straight a below straight x equals straight a equals straight f left parenthesis straight a right parenthesis

⇒ f is continuous for all a > 1
Case II : a < 1
In this case, all nearby points of a are also < 1

Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below straight x squared equals straight a squared equals straight f left parenthesis straight a right parenthesis

⇒ f is continuous for all a < 1.
Case III : a = 1
∴ nearby points can be either <1 or > 1.

space space space space space Lt with straight x rightwards arrow straight a 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 straight x squared space space space space Lt with straight x rightwards arrow straight a 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 straight x equals 1 Also space straight f left parenthesis 1 right parenthesis equals 1 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 equals 1

⇒ f is continuous at x = 1
∴ f is continuous at every point of its domain.

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