Find all points of discontinuity of f, where f is defined by fr
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    Continuity and Differentiability

    Multiple Choice QuestionsLong Answer Type

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    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

    Here space 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
    Function f is defined for all points of the real line.
    Let c be any real number.
    Three cases arise ;
    Case I ; c < 0
    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 fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction equals Lt with straight x rightwards arrow straight c below open parentheses fraction numerator negative straight x over denominator straight x end fraction close parentheses equals Lt with straight x rightwards arrow straight c below left parenthesis negative 1 right parenthesis equals negative 1 Also space space space space space straight f left parenthesis straight c right parenthesis equals fraction numerator open vertical bar straight c close vertical bar over denominator straight c end fraction equals negative straight c over straight c equals negative 1 therefore 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 < 0.
    Case II : c > 0
    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 fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction equals Lt with straight x rightwards arrow straight c below open parentheses straight x over straight x close parentheses equals Lt with straight x rightwards arrow straight c below left parenthesis 1 right parenthesis equals 1 Also space space space space space straight f left parenthesis straight c right parenthesis equals fraction numerator open vertical bar straight c close vertical bar over denominator straight c end fraction equals straight c over straight c equals 1 therefore 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 continuous at all points x > 0
    Case III; c = 0
    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 0 to the power of minus below fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction equals Lt with straight x rightwards arrow 0 to the power of minus below open parentheses fraction numerator negative straight x over denominator straight x end fraction close parentheses equals Lt with straight x rightwards arrow 0 to the power of minus below left parenthesis negative 1 right parenthesis equals negative 1 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 0 to the power of plus below fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below open parentheses straight x over straight x close parentheses equals Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis 1 right parenthesis equals 1 therefore 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 = 0.
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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
    79 Views
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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

    81 Views
    Answer
    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 bystraight 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
    81 Views
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    Multiple Choice QuestionsShort 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
    78 Views
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    Multiple Choice QuestionsLong 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
    80 Views
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    Multiple Choice QuestionsShort Answer Type

    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
    80 Views
    Answer
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