Find all points of discontinuity of f where from Mathematics Co
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    81. If f is continuous and g is a discontinuous function then f + g is discontinuous function.
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    Multiple Choice QuestionsLong Answer Type

    82. Examine the continuity 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 sin space straight x minus cos space straight x comma space space space if space straight x not equal to 0 end cell row cell space space space space space space minus 1 space space space space space space space space space comma space space space if space straight x equals 0 end cell end table close
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    83. Find all points of discontinuity of f where
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space straight x over denominator straight x end fraction comma space space if space straight x less than 0 end cell row cell space space straight x plus 1 comma space space space if space straight x greater or equal than 0 end cell end table close

    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space straight x over denominator straight x end fraction comma space space if space straight x less than 0 end cell row cell space space straight x plus 1 comma space space space if space straight x greater or equal than 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 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 sin space straight x over denominator straight x end fraction equals fraction numerator sin space straight c over denominator straight c end fraction space space space space space space space space straight f left parenthesis straight c right parenthesis equals fraction numerator sin space straight c over denominator straight c end fraction therefore 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 therefore space straight f space is space continous space at space all space points space straight x less than 0.

    Case II:c > 1

    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 left parenthesis straight x plus 1 right parenthesis equals straight c plus 1 space space space space space space straight f left parenthesis straight c right parenthesis equals straight c plus 1 therefore 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 therefore space straight f space is space continous space at space all space points space straight x greater than 0

    Case III: c = 0

    space Lt with straight x rightwards arrow 0 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 sin space straight x over denominator straight x end fraction space space space left square bracket Put space straight x equals 0 minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of minus right square bracket space space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below fraction numerator sin left parenthesis 0 minus straight h right parenthesis over denominator 0 minus straight h end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator sin left parenthesis negative straight h right parenthesis over denominator negative straight h end fraction space space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below fraction numerator negative sin space straight h over denominator negative straight h end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator sin space straight h over denominator straight h end fraction equals 1 therefore space Lt with straight x rightwards arrow 0 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 left parenthesis straight x plus 1 right parenthesis equals 0 plus 1 equals 1 Also space straight f left parenthesis 0 right parenthesis equals 0 plus 1 equals 1 therefore space Lt with straight x rightwards arrow 0 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 plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis
    ∴ f is continuous at x = 0
    Thus f has no point of discontinuity.

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    Multiple Choice QuestionsShort Answer Type

    84. Let f be defined by

    straight f left parenthesis straight x right parenthesis open curly brackets table attributes columnalign left end attributes row cell space space space space space straight x squared over straight x space space space space space space space space space space space space comma space 0 space less or equal than straight x less or equal than 1 end cell row cell 2 straight x squared minus 3 straight x plus 3 over 2 comma space 1 less than straight x less or equal than 2 end cell end table close
    Show that f is continous at x=1.
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    85. Is the function f defined 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 if space straight x less or equal than 1 end cell row cell 5 comma space if space straight x greater than 1 end cell end table close
    continous at x=0? At x=1? At x=2?
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    Multiple Choice QuestionsLong Answer Type

    86. Which of the following functions are continuous at the indicated points ?
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space space space comma space if space straight x less than 0 end cell row cell straight x squared space space comma space if space straight x greater than 0 end cell end table close
    71 Views
    Answer

    Multiple Choice QuestionsShort Answer Type

    87. Which of the following functions are continuous at the indicated points?straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space 5 straight x minus 4 space space space comma space if space straight x less or equal than 1 end cell row cell 4 straight x squared minus 3 straight x space space comma space if space straight x greater than 1 end cell end table close
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    88. Which of the following functions are continuous at the indicated points ?
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space straight x less or equal than 0 end cell row cell 2 straight x plus 1 space space space comma space space space if space straight x greater or equal than 0 end cell end table close
    71 Views
    Answer
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    89. Which of the following functions are continuous at the indicated points ?
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space straight x less than 0 end cell row cell 2 straight x plus 6 space space space comma space space space if space straight x greater than 1 end cell end table close at space straight x equals 0
    87 Views
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    90. Which of the following functions are continuous at the indicated points ?straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space space space straight x less than 2 end cell row cell space space space space space space 3 over 2 space space space space comma space space space if space space space straight x greater or equal than 2 end cell end table close at space straight x equals 2
    72 Views
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