Show that the function of given by from Mathematics Continuity

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Continuity and Differentiability

Long Answer Type

111. 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 fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma space space space space if space straight x not equal to 0 end cell row cell 0 comma space space space space space space space space space if space straight x equals 0 end cell end table close

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112. 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 fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x less than 0 end cell row cell space minus 1 comma space space if space straight x greater or equal than 0 end cell end table close

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113.

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

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114.

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

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115.

Find all points of discontinuity of f, where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x to the power of 10 minus 1 comma space space if space straight x less or equal than 1 end cell row cell straight x squared comma space space space space space space space space space space space if space straight x greater than 1 end cell end table close$

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116. Is the function defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x plus 5 comma space if space straight x less or equal than 1 end cell row cell straight x minus 5 comma space if space straight x greater than 1 end cell end table close

a continuous function?

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117. Discuss the continuity of the function f, where/is defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 3 comma space space if space 0 less or equal than straight x less or equal than 1 space space end cell row cell 4 comma space space if space 1 less than straight x less than 3 end cell row cell 5 comma space if space 3 less or equal than straight x less or equal than 1 space end cell end table close

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Short Answer Type

118. Discuss the continuity of the function f, where f is defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 2 straight x comma end cell row cell 0 comma end cell row cell 4 straight x comma end cell end table close table row cell if space straight x less than 0 space space space space space space end cell row cell if space 0 less or equal than straight x less or equal than 1 end cell row cell if space straight x greater than 1 space space space space space space end cell end table

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Long Answer Type

119. Discuss the continuity of the function f , where fis defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell negative 2 comma space space space space if space straight x less or equal than negative 1 end cell row cell 2 straight x comma space space if minus 1 less than straight x less or equal than 1 end cell row cell 2 space space space space space if space straight x greater than 1 end cell end table close

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Short Answer Type

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

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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